- mathfun@mailman.xmission.com - mailman.xmission.com

[math-fun] emails about div. chains
by N. J. A. Sloane 04 May '04

04 May '04

apparently some people have been posting to the math fun list when they should have been posting to the seqfan list let's keep the sequence stuff on the seq fan list , Ok? and when you have a nice sequence, please send it to the OEIS by filling out the "Submit New Sequence or Comment" web page, OK? Don't expect that i will go though all the postings and all the submissions to try to figure out what was posted and not submitted, and what was submitted and not posted. OK? At the last count i was getting over 350 emails a day. The associate editrors are supposed to help, not make my life more complicated, OK? Here are all the postings about divisor chains that i saved, from both mailing lists. They have been summarized in sequences A094096-A094099. They are not in order and there are duplicates, since they arrived from several sources with multiple copies. I will send this to both lists, to bring everyone up to date unless you are interested in divosor chains, delete this. NJAS >From math-fun-bounces+njas=research.att.com(a)mailman.xmission.com Mon May 3 16:26:28 2004 Received: from mail-gray.research.att.com (mail-gray.research.att.com [135.207.30.104]) by fry.research.att.com (SGI-8.9.3p2/8.8.7) with ESMTP id QAA83757 for <njas(a)fry.research.att.com>; Mon, 3 May 2004 16:26:28 -0400 (EDT) Received: by mail-gray.research.att.com (Postfix) id 9525C75804B; Mon, 3 May 2004 16:25:58 -0400 (EDT) Delivered-To: njas(a)research.att.com Received: by mail-gray.research.att.com (Postfix, from userid 612) id 8BD05758071; Mon, 3 May 2004 16:25:58 -0400 (EDT) Received: from mail-dark.research.att.com (mail-dark.research.att.com [192.20.225.112]) by mail-gray.research.att.com (Postfix) with ESMTP id 74F8375804B for <njas(a)research.att.com>; Mon, 3 May 2004 16:25:57 -0400 (EDT) Received: from mailman.xmission.com (mailman.xmission.com [198.60.22.29]) by mail-dark.research.att.com (Postfix) with ESMTP id 46546C8328 for <njas(a)research.att.com>; Mon, 3 May 2004 16:25:57 -0400 (EDT) Received: from localhost ([127.0.0.1] helo=mailman.xmission.com) by mailman.xmission.com with esmtp (Exim 3.36 #1 (Debian)) id 1BKk0i-0001yQ-06 for <njas(a)research.att.com>; Mon, 03 May 2004 14:25:56 -0600 Received: from mgr5.xmission.com ([198.60.22.205]) by mailman.xmission.com with esmtp (Exim 3.36 #1 (Debian)) id 1BKk0g-0001yJ-00 for <math-fun(a)mailman.xmission.com>; Mon, 03 May 2004 14:25:54 -0600 Received: from [136.159.2.1] (helo=ensa.cpsc.ucalgary.ca) by mgr5.xmission.com with esmtp (Exim 4.30) id 1BKk0c-0004k7-QS for math-fun(a)mailman.xmission.com; Mon, 03 May 2004 14:25:50 -0600 Received: from imgw1.cpsc.ucalgary.ca (imgw1 [136.159.5.9]) by ensa.cpsc.ucalgary.ca (8.12.10/8.12.10) with ESMTP id i43KMZ4b026878; Mon, 3 May 2004 14:22:35 -0600 (MDT) Received: from csl (csl [136.159.5.22]) by imgw1.cpsc.ucalgary.ca (8.12.10/8.12.10) with ESMTP id i43KMSC3019488; Mon, 3 May 2004 14:22:28 -0600 Date: Mon, 3 May 2004 14:20:45 -0600 (MDT) From: Richard Guy <rkg(a)cpsc.ucalgary.ca> X-X-Sender: rkg@csl To: Math Fun <math-fun(a)mailman.xmission.com>, <seqfan(a)ext.jussieu.fr> Message-ID: <Pine.LNX.4.44.0405031403560.12751-100000@csl> MIME-Version: 1.0 X-Virus-Scanned: by amavis-milter (http://amavis.org/) Content-Type: TEXT/PLAIN; charset=US-ASCII X-SA-Exim-Mail-From: rkg(a)cpsc.ucalgary.ca X-SA-Exim-Version: 3.1 (built Mon Jan 26 13:00:24 MST 2004) X-SA-Exim-Scanned: Yes Cc: Paul Vaderlind <paul(a)math.su.se>, "Vaderlind, Paul -- Paul Vaderlind" <paul(a)matematik.su.se> Subject: [math-fun] Divisor chains X-BeenThere: math-fun(a)mailman.xmission.com X-Mailman-Version: 2.1.4 Precedence: list Reply-To: math-fun <math-fun(a)mailman.xmission.com> List-Id: math-fun <math-fun.mailman.xmission.com> List-Unsubscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=unsubscribe> List-Archive: <http://mailman.xmission.com/cgi-bin/mailman/private/math-fun> List-Post: <mailto:math-fun@mailman.xmission.com> List-Help: <mailto:math-fun-request@mailman.xmission.com?subject=help> List-Subscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=subscribe> Sender: math-fun-bounces+njas=research.att.com(a)mailman.xmission.com Errors-To: math-fun-bounces+njas=research.att.com(a)mailman.xmission.com X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on mail-gray.research.att.com X-Spam-Level: X-Spam-Status: No, hits=-104.9 required=4.0 tests=AWL,BAYES_00, USER_IN_ALL_SPAM_TO autolearn=ham version=2.63 Status: R The following might be an acceptable sequence for Neil Sloane's OEIS, but someone needs to do some work 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ... 1 1 1 1 1 1 1 5 4 1 2 5 5 4 ... It arises from a problem I got recently from Paul Vaderlind. If the sequence of numbers from 1 to 37, is arranged so that each term is a divisor of the sum of preceding ones, starting 37, 1, ... what is the next term? The answer is either 2 or 19. P'r'aps I won't spoil your fun by pointing out which. But I will spoil it by asking: How do you know that there is such a `divisor chain' ? The sequence is (my present state of knowledge of) the number of divisor chains of length n. Here are the ones I found 1 2 1 3 1 2 4 2 3 1 5 1 2 4 3 6 2 4 3 5 1 7 1 2 5 3 6 4 8 2 5 3 6 4 7 1 8 4 2 7 3 1 5 6 8 4 2 7 3 6 5 1 8 4 3 5 1 7 2 6 8 4 6 3 7 2 5 1 9 1 2 4 8 6 5 7 3 9 1 2 6 3 7 4 8 5 9 3 4 8 6 5 7 2 1 9 3 6 2 1 7 4 8 5 10 2 4 8 3 9 6 7 1 5 11 1 2 7 3 8 4 9 5 10 6 11 1 4 8 6 10 5 9 2 7 3 12 2 1 5 10 3 11 4 8 7 9 6 12 2 7 3 8 4 9 5 10 6 11 1 12 3 5 10 2 8 4 11 1 7 9 6 12 4 8 3 9 6 2 11 5 10 7 1 12 4 8 6 10 5 9 2 7 3 11 1 13 1 2 8 3 9 4 10 5 11 6 12 7 13 1 2 8 6 10 4 11 5 12 9 3 7 13 1 2 8 6 10 4 11 5 3 9 12 7 13 1 2 8 12 4 10 5 11 3 9 3 7 13 1 2 8 12 4 10 5 11 6 9 3 7 14 2 8 6 10 4 11 5 3 9 12 7 13 1 14 2 8 6 10 4 11 5 12 9 3 7 13 1 14 2 8 12 4 10 5 11 6 9 3 7 13 1 14 2 8 12 9 3 4 13 1 11 7 6 10 5 Enough sins of omission, to say nothing of commission, for today. Please check and extend. Any ideas for proving anything? R. _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >From math-fun-bounces+njas=research.att.com(a)mailman.xmission.com Mon May 3 16:53:57 2004 Received: from mail-brown.research.att.com (mail-brown.research.att.com [135.207.30.105]) by fry.research.att.com (SGI-8.9.3p2/8.8.7) with ESMTP id QAA45551 for <njas(a)fry.research.att.com>; Mon, 3 May 2004 16:53:57 -0400 (EDT) Received: by mail-brown.research.att.com (Postfix) id CEC232D40A3; Mon, 3 May 2004 16:53:27 -0400 (EDT) Delivered-To: njas(a)research.att.com Received: by mail-brown.research.att.com (Postfix, from userid 612) id C0C982D40A5; Mon, 3 May 2004 16:53:27 -0400 (EDT) Received: from mail-dark.research.att.com (mail-dark.research.att.com [192.20.225.112]) by mail-brown.research.att.com (Postfix) with ESMTP id 95AA82D40A3 for <njas(a)research.att.com>; Mon, 3 May 2004 16:53:26 -0400 (EDT) Received: from mailman.xmission.com (mailman.xmission.com [198.60.22.29]) by mail-dark.research.att.com (Postfix) with ESMTP id 6A84FC829E for <njas(a)research.att.com>; Mon, 3 May 2004 16:53:26 -0400 (EDT) Received: from localhost ([127.0.0.1] helo=mailman.xmission.com) by mailman.xmission.com with esmtp (Exim 3.36 #1 (Debian)) id 1BKkRK-0005q9-00 for <njas(a)research.att.com>; Mon, 03 May 2004 14:53:26 -0600 Received: from mgr8.xmission.com ([198.60.22.208] ident=mail) by mailman.xmission.com with esmtp (Exim 3.36 #1 (Debian)) id 1BKkRH-0005q2-00 for <math-fun(a)mailman.xmission.com>; Mon, 03 May 2004 14:53:23 -0600 Received: from [66.218.78.168] (helo=web40711.mail.yahoo.com) by mgr8.xmission.com with smtp (Exim 4.30) id 1BKkRE-0002aQ-SN for math-fun(a)mailman.xmission.com; Mon, 03 May 2004 14:53:20 -0600 Message-ID: <20040503205320.41955.qmail(a)web40711.mail.yahoo.com> Received: from [65.206.211.14] by web40711.mail.yahoo.com via HTTP; Mon, 03 May 2004 13:53:20 PDT Date: Mon, 3 May 2004 13:53:20 -0700 (PDT) From: Eugene Salamin <gene_salamin(a)yahoo.com> To: math-fun <math-fun(a)mailman.xmission.com> In-Reply-To: <Pine.LNX.4.44.0405031403560.12751-100000@csl> MIME-Version: 1.0 Subject: Re: [math-fun] Divisor chains Content-Type: text/plain; charset=us-ascii X-SA-Exim-Mail-From: gene_salamin(a)yahoo.com X-SA-Exim-Version: 3.1 (built Mon Jan 26 13:00:24 MST 2004) X-SA-Exim-Scanned: Yes X-BeenThere: math-fun(a)mailman.xmission.com X-Mailman-Version: 2.1.4 Precedence: list Reply-To: math-fun <math-fun(a)mailman.xmission.com> List-Id: math-fun <math-fun.mailman.xmission.com> List-Unsubscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=unsubscribe> List-Archive: <http://mailman.xmission.com/cgi-bin/mailman/private/math-fun> List-Post: <mailto:math-fun@mailman.xmission.com> List-Help: <mailto:math-fun-request@mailman.xmission.com?subject=help> List-Subscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=subscribe> Sender: math-fun-bounces+njas=research.att.com(a)mailman.xmission.com Errors-To: math-fun-bounces+njas=research.att.com(a)mailman.xmission.com X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on mail-brown.research.att.com X-Spam-Level: X-Spam-Status: No, hits=-104.9 required=4.0 tests=AWL,BAYES_00, USER_IN_ALL_SPAM_TO autolearn=ham version=2.63 Status: R You seem to be considering only those chains for which the first element is the largest. Otherwise we have chains such as (2 1 3), (2 4 3 1), (2 4 3 1 5), (3 1 4 2 5), or (4 1 5 2 3). Gene --- Richard Guy <rkg(a)cpsc.ucalgary.ca> wrote: > The following might be an acceptable sequence > for Neil Sloane's OEIS, but someone needs to > do some work > > 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ... > 1 1 1 1 1 1 1 5 4 1 2 5 5 4 ... > > It arises from a problem I got recently from > Paul Vaderlind. > > If the sequence of numbers from 1 to 37, > is arranged so that each term is a divisor > of the sum of preceding ones, starting > 37, 1, ... > what is the next term? > > The answer is either 2 or 19. P'r'aps > I won't spoil your fun by pointing out > which. But I will spoil it by asking: > How do you know that there is such a > `divisor chain' ? > > The sequence is (my present state of knowledge > of) the number of divisor chains of length n. > Here are the ones I found > > 1 2 1 3 1 2 4 2 3 1 > > 5 1 2 4 3 6 2 4 3 5 1 7 1 2 5 3 6 4 > > 8 2 5 3 6 4 7 1 > 8 4 2 7 3 1 5 6 > 8 4 2 7 3 6 5 1 > 8 4 3 5 1 7 2 6 > 8 4 6 3 7 2 5 1 > > 9 1 2 4 8 6 5 7 3 > 9 1 2 6 3 7 4 8 5 > 9 3 4 8 6 5 7 2 1 > 9 3 6 2 1 7 4 8 5 > > 10 2 4 8 3 9 6 7 1 5 > > 11 1 2 7 3 8 4 9 5 10 6 > 11 1 4 8 6 10 5 9 2 7 3 > > 12 2 1 5 10 3 11 4 8 7 9 6 > 12 2 7 3 8 4 9 5 10 6 11 1 > 12 3 5 10 2 8 4 11 1 7 9 6 > 12 4 8 3 9 6 2 11 5 10 7 1 > 12 4 8 6 10 5 9 2 7 3 11 1 > > 13 1 2 8 3 9 4 10 5 11 6 12 7 > 13 1 2 8 6 10 4 11 5 12 9 3 7 > 13 1 2 8 6 10 4 11 5 3 9 12 7 > 13 1 2 8 12 4 10 5 11 3 9 3 7 > 13 1 2 8 12 4 10 5 11 6 9 3 7 > > 14 2 8 6 10 4 11 5 3 9 12 7 13 1 > 14 2 8 6 10 4 11 5 12 9 3 7 13 1 > 14 2 8 12 4 10 5 11 6 9 3 7 13 1 > 14 2 8 12 9 3 4 13 1 11 7 6 10 5 > > Enough sins of omission, to say nothing of > commission, for today. Please check and extend. > Any ideas for proving anything? R. > > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun __________________________________ Do you Yahoo!? Win a $20,000 Career Makeover at Yahoo! HotJobs http://hotjobs.sweepstakes.yahoo.com/careermakeover _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >From math-fun-bounces+njas=research.att.com(a)mailman.xmission.com Tue May 4 02:03:52 2004 Received: from mail-brown.research.att.com (mail-brown.research.att.com [135.207.30.105]) by fry.research.att.com (SGI-8.9.3p2/8.8.7) with ESMTP id CAA80448 for <njas(a)fry.research.att.com>; Tue, 4 May 2004 02:03:52 -0400 (EDT) Received: by mail-brown.research.att.com (Postfix) id 62AD72D40C9; Tue, 4 May 2004 02:03:22 -0400 (EDT) Delivered-To: njas(a)research.att.com Received: by mail-brown.research.att.com (Postfix, from userid 612) id 5642A2D40C8; Tue, 4 May 2004 02:03:22 -0400 (EDT) Received: from mail-dark.research.att.com (mail-dark.research.att.com [192.20.225.112]) by mail-brown.research.att.com (Postfix) with ESMTP id 1E08D2D40C6 for <njas(a)research.att.com>; Tue, 4 May 2004 02:03:21 -0400 (EDT) Received: from mailman.xmission.com (mailman.xmission.com [198.60.22.29]) by mail-dark.research.att.com (Postfix) with ESMTP id EA5DBC84BF for <njas(a)research.att.com>; Tue, 4 May 2004 02:03:20 -0400 (EDT) Received: from localhost ([127.0.0.1] helo=mailman.xmission.com) by mailman.xmission.com with esmtp (Exim 3.36 #1 (Debian)) id 1BKt1U-0007b9-06 for <njas(a)research.att.com>; Tue, 04 May 2004 00:03:20 -0600 Received: from mgr9.xmission.com ([198.60.22.209]) by mailman.xmission.com with esmtp (Exim 3.36 #1 (Debian)) id 1BKt1R-0007aq-00 for <math-fun(a)mailman.xmission.com>; Tue, 04 May 2004 00:03:17 -0600 Received: from [130.221.88.102] (helo=mhultra.aero.org) by mgr9.xmission.com with esmtp (Exim 4.30) id 1BKt1P-0003fJ-Sk for math-fun(a)mailman.xmission.com; Tue, 04 May 2004 00:03:15 -0600 Received: from rushe.aero.org ([130.221.24.10] [130.221.24.10]) by mhultra.aero.org with ESMTP for math-fun(a)mailman.xmission.com; Mon, 3 May 2004 23:02:44 -0700 Received: from calamari.aero.org (calamari.aero.org [130.221.26.26]) by rushe.aero.org (8.11.7p1+Sun/8.11.6) with ESMTP id i4462hD06243; Mon, 3 May 2004 23:02:43 -0700 (PDT) From: Chris Landauer <cal(a)rushg.aero.org> Received: (from cal@localhost) by calamari.aero.org (8.11.6/8.11.6) id i4465Ko96093; Mon, 3 May 2004 23:05:20 -0700 (PDT) (envelope-from cal) Date: Mon, 3 May 2004 23:05:20 -0700 (PDT) Message-Id: <200405040605.i4465Ko96093(a)calamari.aero.org> To: math-fun(a)mailman.xmission.com X-SA-Exim-Mail-From: cal(a)rushg.aero.org X-SA-Exim-Version: 3.1 (built Mon Jan 26 13:00:24 MST 2004) X-SA-Exim-Scanned: Yes Cc: cal(a)rushg.aero.org Subject: [math-fun] penultimate divisor counts output X-BeenThere: math-fun(a)mailman.xmission.com X-Mailman-Version: 2.1.4 Precedence: list Reply-To: math-fun <math-fun(a)mailman.xmission.com> List-Id: math-fun <math-fun.mailman.xmission.com> List-Unsubscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=unsubscribe> List-Archive: <http://mailman.xmission.com/cgi-bin/mailman/private/math-fun> List-Post: <mailto:math-fun@mailman.xmission.com> List-Help: <mailto:math-fun-request@mailman.xmission.com?subject=help> List-Subscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=subscribe> Sender: math-fun-bounces+njas=research.att.com(a)mailman.xmission.com Errors-To: math-fun-bounces+njas=research.att.com(a)mailman.xmission.com X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on mail-brown.research.att.com X-Spam-Level: X-Spam-Status: No, hits=-104.9 required=4.0 tests=AWL,BAYES_00, USER_IN_ALL_SPAM_TO autolearn=ham version=2.63 Status: R hihi, all - yes, well, there were some typos - now the program counts them, so i don't have to there will be a few more numbers tomorrow (hopefully, n=37 will appear), but for now, here is what i have 1 chains of length 1, 1 anchored, 1 cyclic, 1 both 1 chains of length 2, 1 anchored, 0 cyclic, 0 both 2 chains of length 3, 1 anchored, 2 cyclic, 1 both 2 chains of length 4, 1 anchored, 0 cyclic, 0 both 4 chains of length 5, 1 anchored, 2 cyclic, 1 both 5 chains of length 6, 1 anchored, 0 cyclic, 0 both 7 chains of length 7, 1 anchored, 3 cyclic, 1 both 7 chains of length 8, 5 anchored, 0 cyclic, 0 both 24 chains of length 9, 4 anchored, 5 cyclic, 4 both 22 chains of length 10, 3 anchored, 0 cyclic, 0 both 29 chains of length 11, 2 anchored, 6 cyclic, 2 both 39 chains of length 12, 8 anchored, 0 cyclic, 0 both 67 chains of length 13, 4 anchored, 6 cyclic, 4 both 55 chains of length 14, 6 anchored, 0 cyclic, 0 both 386 chains of length 15, 47 anchored, 147 cyclic, 47 both 235 chains of length 16, 44 anchored, 1 cyclic, 0 both 312 chains of length 17, 6 anchored, 22 cyclic, 6 both 347 chains of length 18, 37 anchored, 2 cyclic, 0 both 451 chains of length 19, 6 anchored, 27 cyclic, 6 both 1319 chains of length 20, 166 anchored, 165 cyclic, 0 both 5320 chains of length 21, 462 anchored, 519 cyclic, 462 both 3220 chains of length 22, 232 anchored, 0 cyclic, 0 both 4489 chains of length 23, 372 anchored, 516 cyclic, 372 both 20237 chains of length 24, 2130 anchored, 2021 cyclic, 0 both 36580 chains of length 25, 1589 anchored, 1912 cyclic, 1589 both 52875 chains of length 26, 9093 anchored, 506 cyclic, 0 both 197103 chains of length 27, 20896 anchored, 45658 cyclic, 20896 both 216562 chains of length 28, 20314 anchored, 514 cyclic, 0 both (for a sequence to be both cyclic and anchored, the first element must both divide n*(n+1)/2 and be equal to n, so n must be odd) one more tomorrow, cal Chris Landauer Aerospace Integration Science Center The Aerospace Corporation cal(a)aero.org _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >From seqfan-owner(a)ext.jussieu.fr Tue May 4 03:32:32 2004 Received: from mail-brown.research.att.com (mail-brown.research.att.com [135.207.30.105]) by fry.research.att.com (SGI-8.9.3p2/8.8.7) with ESMTP id DAA52697 for <njas(a)fry.research.att.com>; Tue, 4 May 2004 03:32:32 -0400 (EDT) Received: by mail-brown.research.att.com (Postfix) id 75C7D2D40BE; Tue, 4 May 2004 03:32:02 -0400 (EDT) Delivered-To: njas(a)research.att.com Received: by mail-brown.research.att.com (Postfix, from userid 612) id 6975D2D40C6; Tue, 4 May 2004 03:32:02 -0400 (EDT) Received: from mail-dark.research.att.com (mail-dark.research.att.com [192.20.225.112]) by mail-brown.research.att.com (Postfix) with ESMTP id 92E002D40BE for <njas(a)research.att.com>; Tue, 4 May 2004 03:32:01 -0400 (EDT) Received: from shiva.jussieu.fr (shiva.jussieu.fr [134.157.0.129]) by mail-dark.research.att.com (Postfix) with ESMTP id A0C53C85A0 for <njas(a)research.att.com>; Tue, 4 May 2004 03:31:59 -0400 (EDT) Received: from idf.ext.jussieu.fr (idf.ext.jussieu.fr [134.157.81.129]) by shiva.jussieu.fr (8.12.11/jtpda-5.4) with ESMTP id i447TF9t097994 ; Tue, 4 May 2004 09:29:15 +0200 (CEST) Received: from idf.ext.jussieu.fr (localhost [127.0.0.1]) by idf.ext.jussieu.fr (8.12.6/jtpda-5.4) with ESMTP id i447TvUk098840 ; Tue, 4 May 2004 09:29:57 +0200 (CEST) Received: (from sympa@localhost) by idf.ext.jussieu.fr (8.12.6/8.12.6/Submit) id i447TudK098835; Tue, 4 May 2004 09:29:56 +0200 (CEST) X-Authentication-Warning: idf.ext.jussieu.fr: sympa set sender to seqfan-owner(a)ext.jussieu.fr using -f Received: from shiva.jussieu.fr (shiva.jussieu.fr [134.157.0.129]) by idf.ext.jussieu.fr (8.12.6/jtpda-5.4) with ESMTP id i447TqUk098810 for <seqfan(a)ext.jussieu.fr>; Tue, 4 May 2004 09:29:52 +0200 (CEST) Received: from goose.mail.pas.earthlink.net (goose.mail.pas.earthlink.net [207.217.120.18]) by shiva.jussieu.fr (8.12.11/jtpda-5.4) with ESMTP id i447Te1Z023110 for <seqfan(a)ext.jussieu.fr>; Tue, 4 May 2004 09:29:46 +0200 (CEST) X-Ids: 165 Received: from pool0061.cvx22-bradley.dialup.earthlink.net ([209.179.198.61] helo=[209.179.198.66]) by goose.mail.pas.earthlink.net with esmtp (Exim 3.33 #1) id 1BKuN6-0002Kq-00 for seqfan(a)ext.jussieu.fr; Tue, 04 May 2004 00:29:44 -0700 Mime-Version: 1.0 X-Sender: dwasserm(a)pop.earthlink.net Message-Id: <v04210102bcbcf5ae537d(a)[209.179.198.66]> In-Reply-To: <Pine.LNX.4.44.0405031403560.12751-100000@csl> References: <Pine.LNX.4.44.0405031403560.12751-100000@csl> X-mailer: Eudora Pro 4.2.1 Macintosh Date: Tue, 4 May 2004 00:30:16 -0700 To: seqfan(a)ext.jussieu.fr From: David Wasserman <dwasserm(a)earthlink.com> Subject: Re: Divisor chains Content-Type: text/plain; charset="us-ascii" X-j-chkmail-Score: MSGID : 40974664.000 on shiva.jussieu.fr : j-chkmail score : X : 0/50 1 X-Miltered: at shiva.jussieu.fr with ID 4097464B.001 by Joe's j-chkmail (http://j-chkmail.ensmp.fr) X-Miltered: at shiva.jussieu.fr with ID 40974664.000 by Joe's j-chkmail (http://j-chkmail.ensmp.fr) X-Antivirus: scanned by sophie at shiva.jussieu.fr X-Antivirus: scanned by sophie at shiva.jussieu.fr X-Loop: seqfan(a)ext.jussieu.fr X-Sequence: 3957 Errors-To: seqfan-owner(a)ext.jussieu.fr Precedence: list X-no-archive: yes List-Id: <seqfan.ext.jussieu.fr> List-Help: <mailto:sympa@ext.jussieu.fr?subject=help> List-Subscribe: <mailto:sympa@ext.jussieu.fr?subject=subscribe%20seqfan> List-Unsubscribe: <mailto:sympa@ext.jussieu.fr?subject=unsubscribe%20seqfan> List-Post: <mailto:seqfan@ext.jussieu.fr> List-Owner: <mailto:seqfan-request@ext.jussieu.fr> X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on mail-brown.research.att.com X-Spam-Level: X-Spam-Status: No, hits=-92.3 required=4.0 tests=AWL,BAYES_00, USER_IN_ALL_SPAM_TO autolearn=ham version=2.63 Status: R >From the results that have been posted, I expect that there are many divisor chains for n = 37, but no one has exhibited any. So here's two of them: 37 1 2 4 11 5 3 7 14 6 15 35 28 8 22 33 21 36 18 34 20 12 31 13 32 16 29 17 30 10 25 23 26 24 27 9 19 37 1 2 4 11 5 3 7 14 6 15 35 28 8 22 33 21 36 18 34 20 12 31 13 32 16 29 17 30 10 25 23 26 24 27 9 19 I found these in about 20 minutes of computer time. >From math-fun-bounces+njas=research.att.com(a)mailman.xmission.com Tue May 4 03:48:25 2004 Received: from mail-gray.research.att.com (mail-gray.research.att.com [135.207.30.104]) by fry.research.att.com (SGI-8.9.3p2/8.8.7) with ESMTP id DAA30738 for <njas(a)fry.research.att.com>; Tue, 4 May 2004 03:48:25 -0400 (EDT) Received: by mail-gray.research.att.com (Postfix) id 37AF7758085; Tue, 4 May 2004 03:47:55 -0400 (EDT) Delivered-To: njas(a)research.att.com Received: by mail-gray.research.att.com (Postfix, from userid 612) id 2873E758086; Tue, 4 May 2004 03:47:55 -0400 (EDT) Received: from mail-dark.research.att.com (mail-dark.research.att.com [192.20.225.112]) by mail-gray.research.att.com (Postfix) with ESMTP id 02D6D758085 for <njas(a)research.att.com>; Tue, 4 May 2004 03:47:54 -0400 (EDT) Received: from mailman.xmission.com (mailman.xmission.com [198.60.22.29]) by mail-dark.research.att.com (Postfix) with ESMTP id CB7B7C865E for <njas(a)research.att.com>; Tue, 4 May 2004 03:47:53 -0400 (EDT) Received: from localhost ([127.0.0.1] helo=mailman.xmission.com) by mailman.xmission.com with esmtp (Exim 3.36 #1 (Debian)) id 1BKuef-0006pL-01 for <njas(a)research.att.com>; Tue, 04 May 2004 01:47:53 -0600 Received: from mgr8.xmission.com ([198.60.22.208] ident=mail) by mailman.xmission.com with esmtp (Exim 3.36 #1 (Debian)) id 1BKueb-0006pE-00 for <math-fun(a)mailman.xmission.com>; Tue, 04 May 2004 01:47:49 -0600 Received: from [207.217.120.50] (helo=avocet.mail.pas.earthlink.net) by mgr8.xmission.com with esmtp (Exim 4.30) id 1BKueZ-0003ap-Rm for math-fun(a)mailman.xmission.com; Tue, 04 May 2004 01:47:47 -0600 Received: from daisy.psp.pas.earthlink.net ([207.217.78.223]) by avocet.mail.pas.earthlink.net with esmtp (Exim 3.33 #1) id 1BKueY-00017U-00; Tue, 04 May 2004 00:47:46 -0700 Message-ID: <13375876.1083656866205.JavaMail.root(a)daisy.psp.pas.earthlink.net> Date: Tue, 4 May 2004 00:47:45 -0700 (GMT-07:00) From: Daniel Asimov <dasimov(a)earthlink.net> To: math-fun <math-fun(a)mailman.xmission.com>, seqfan(a)ext.jussieu.fr Mime-Version: 1.0 X-Mailer: Earthlink Zoo Mail 1.0 Subject: Re: [math-fun] Divisor chains Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-SA-Exim-Mail-From: dasimov(a)earthlink.net X-SA-Exim-Version: 3.1 (built Mon Jan 26 13:00:24 MST 2004) X-SA-Exim-Scanned: Yes Cc: Paul Vaderlind <paul(a)math.su.se> X-BeenThere: math-fun(a)mailman.xmission.com X-Mailman-Version: 2.1.4 Precedence: list Reply-To: Daniel Asimov <dasimov(a)earthlink.net>, math-fun <math-fun(a)mailman.xmission.com> List-Id: math-fun <math-fun.mailman.xmission.com> List-Unsubscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=unsubscribe> List-Archive: <http://mailman.xmission.com/cgi-bin/mailman/private/math-fun> List-Post: <mailto:math-fun@mailman.xmission.com> List-Help: <mailto:math-fun-request@mailman.xmission.com?subject=help> List-Subscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=subscribe> Sender: math-fun-bounces+njas=research.att.com(a)mailman.xmission.com Errors-To: math-fun-bounces+njas=research.att.com(a)mailman.xmission.com X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on mail-gray.research.att.com X-Spam-Level: X-Spam-Status: No, hits=-104.9 required=4.0 tests=BAYES_00, USER_IN_ALL_SPAM_TO autolearn=ham version=2.63 Status: R RIchard Guy wrote: << How do you know that there is such a `divisor chain' ? It took quite a while for me to realize that for {1,2,...,n}, if we let M = floor(n/2), the following permutations always work: For n even: M+1, 1, M+2, 2, ..., n-1, M-1, n, M. For n odd: M+1, 1, M+2, 2, ..., M-1, n-1, M, n The proof is both magical and completely obvious. E.g., for n = 10: 6, 1, 7, 2, 8, 3, 9, 4, 10, 5; for n = 37: 19, 1, 20, 2, ..., 35, 17, 36, 18, 37. Still don't know how to show there exists a 37-chain beginning with 37, 1. But since the final term must divide 37*19, the third term must be 2 in any such chain. D'uh! --Dan _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >From seqfan-owner(a)ext.jussieu.fr Tue May 4 06:16:45 2004 Received: from mail-gray.research.att.com (mail-gray.research.att.com [135.207.30.104]) by fry.research.att.com (SGI-8.9.3p2/8.8.7) with ESMTP id GAA62813 for <njas(a)fry.research.att.com>; Tue, 4 May 2004 06:16:45 -0400 (EDT) Received: by mail-gray.research.att.com (Postfix) id E2FAD758088; Tue, 4 May 2004 06:16:14 -0400 (EDT) Delivered-To: njas(a)research.att.com Received: by mail-gray.research.att.com (Postfix, from userid 612) id D4DDB75808B; Tue, 4 May 2004 06:16:14 -0400 (EDT) Received: from mail-dark.research.att.com (mail-dark.research.att.com [192.20.225.112]) by mail-gray.research.att.com (Postfix) with ESMTP id B369B758088 for <njas(a)research.att.com>; Tue, 4 May 2004 06:16:13 -0400 (EDT) Received: from shiva.jussieu.fr (shiva.jussieu.fr [134.157.0.129]) by mail-dark.research.att.com (Postfix) with ESMTP id D0EAFC814C for <njas(a)research.att.com>; Tue, 4 May 2004 06:16:11 -0400 (EDT) Received: from idf.ext.jussieu.fr (idf.ext.jussieu.fr [134.157.81.129]) by shiva.jussieu.fr (8.12.11/jtpda-5.4) with ESMTP id i44AEdA6063258 ; Tue, 4 May 2004 12:14:39 +0200 (CEST) Received: from idf.ext.jussieu.fr (localhost [127.0.0.1]) by idf.ext.jussieu.fr (8.12.6/jtpda-5.4) with ESMTP id i44AFKUk023611 ; Tue, 4 May 2004 12:15:20 +0200 (CEST) Received: (from sympa@localhost) by idf.ext.jussieu.fr (8.12.6/8.12.6/Submit) id i44AEwf3023404; Tue, 4 May 2004 12:14:58 +0200 (CEST) X-Authentication-Warning: idf.ext.jussieu.fr: sympa set sender to seqfan-owner(a)ext.jussieu.fr using -f Received: from shiva.jussieu.fr (shiva.jussieu.fr [134.157.0.129]) by idf.ext.jussieu.fr (8.12.6/jtpda-5.4) with ESMTP id i44AEqUk023377 for <seqfan(a)ext.jussieu.fr>; Tue, 4 May 2004 12:14:52 +0200 (CEST) Received: from mtiwmhc13.worldnet.att.net (mtiwmhc13.worldnet.att.net [204.127.131.117]) by shiva.jussieu.fr (8.12.11/jtpda-5.4) with ESMTP id i44AE2r7062560 for <seqfan(a)ext.jussieu.fr>; Tue, 4 May 2004 12:14:09 +0200 (CEST) X-Ids: 168 Received: from plastereddragon (4.cambridge-10rh15rt.ma.dial-access.att.net[12.76.185.4]) by worldnet.att.net (mtiwmhc13) with SMTP id <2004050410144311300fsl2te>; Tue, 4 May 2004 10:14:43 +0000 Message-ID: <000f01c431c1$7e4dbed0$0200a8c0@plastereddragon> Reply-To: "Chuck Seggelin" <seqfan(a)plastereddragon.com> From: "Chuck Seggelin" <seqfan(a)plastereddragon.com> To: <seqfan(a)ext.jussieu.fr> References: <Pine.LNX.4.44.0405031403560.12751-100000@csl> <v04210102bcbcf5ae537d(a)[209.179.198.66]> Subject: Re: [seqfan] Re: Divisor chains Date: Tue, 4 May 2004 06:20:57 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2800.1106 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2800.1106 X-Miltered: at shiva.jussieu.fr with ID 40976D0F.000 by Joe's j-chkmail (http://j-chkmail.ensmp.fr) X-Miltered: at shiva.jussieu.fr with ID 40976CEA.003 by Joe's j-chkmail (http://j-chkmail.ensmp.fr) X-Antivirus: scanned by sophie at shiva.jussieu.fr X-Antivirus: scanned by sophie at shiva.jussieu.fr X-Loop: seqfan(a)ext.jussieu.fr X-Sequence: 3958 Errors-To: seqfan-owner(a)ext.jussieu.fr Precedence: list X-no-archive: yes List-Id: <seqfan.ext.jussieu.fr> List-Help: <mailto:sympa@ext.jussieu.fr?subject=help> List-Subscribe: <mailto:sympa@ext.jussieu.fr?subject=subscribe%20seqfan> List-Unsubscribe: <mailto:sympa@ext.jussieu.fr?subject=unsubscribe%20seqfan> List-Post: <mailto:seqfan@ext.jussieu.fr> List-Owner: <mailto:seqfan-request@ext.jussieu.fr> X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on mail-gray.research.att.com X-Spam-Level: X-Spam-Status: No, hits=-104.3 required=4.0 tests=AWL,BAYES_00, USER_IN_ALL_SPAM_TO autolearn=ham version=2.63 Status: R In about 6 hours of computing time, I've got 2283 chains for n=37 so far and clearly many more to go. There are 1572 chains that begin with 37,1,2,4,11. -- Chuck ----- Original Message ----- From: "David Wasserman" <dwasserm(a)earthlink.com> To: <seqfan(a)ext.jussieu.fr> Sent: Tuesday, May 04, 2004 3:30 AM Subject: [seqfan] Re: Divisor chains > From the results that have been posted, I expect that there are many divisor chains for n = 37, but no one has exhibited any. So here's two of them: > > 37 1 2 4 11 5 3 7 14 6 15 35 28 8 22 33 21 36 18 34 20 12 31 13 32 16 29 17 30 10 25 23 26 24 27 9 19 > > 37 1 2 4 11 5 3 7 14 6 15 35 28 8 22 33 21 36 18 34 20 12 31 13 32 16 29 17 30 10 25 23 26 24 27 9 19 > > I found these in about 20 minutes of computer time. > Received: from mail-gray.research.att.com (mail-gray.research.att.com [135.207.30.104]) by fry.research.att.com (SGI-8.9.3p2/8.8.7) with ESMTP id TAA05125 for <njas(a)fry.research.att.com>; Mon, 3 May 2004 19:24:20 -0400 (EDT) Received: by mail-gray.research.att.com (Postfix) id E722A758019; Mon, 3 May 2004 19:23:49 -0400 (EDT) Delivered-To: njas(a)research.att.com Received: by mail-gray.research.att.com (Postfix, from userid 612) id D963675805B; Mon, 3 May 2004 19:23:49 -0400 (EDT) Received: from mail-dark.research.att.com (mail-dark.research.att.com [192.20.225.112]) by mail-gray.research.att.com (Postfix) with ESMTP id 9F783758019 for <njas(a)research.att.com>; Mon, 3 May 2004 19:23:48 -0400 (EDT) Received: from mailman.xmission.com (mailman.xmission.com [198.60.22.29]) by mail-dark.research.att.com (Postfix) with ESMTP id 767F2C8130 for <njas(a)research.att.com>; Mon, 3 May 2004 19:23:48 -0400 (EDT) Received: from localhost ([127.0.0.1] helo=mailman.xmission.com) by mailman.xmission.com with esmtp (Exim 3.36 #1 (Debian)) id 1BKmmp-0003j3-0B for <njas(a)research.att.com>; Mon, 03 May 2004 17:23:47 -0600 Received: from mgr7.xmission.com ([198.60.22.207] ident=mail) by mailman.xmission.com with esmtp (Exim 3.36 #1 (Debian)) id 1BKmmn-0003iw-00 for <math-fun(a)mailman.xmission.com>; Mon, 03 May 2004 17:23:45 -0600 Received: from [130.221.88.102] (helo=mhultra.aero.org) by mgr7.xmission.com with esmtp (Exim 4.30) id 1BKmmm-00028l-Ra for math-fun(a)mailman.xmission.com; Mon, 03 May 2004 17:23:44 -0600 Received: from rushe.aero.org ([130.221.24.10] [130.221.24.10]) by mhultra.aero.org with ESMTP for math-fun(a)mailman.xmission.com; Mon, 3 May 2004 16:22:26 -0700 Received: from calamari.aero.org (calamari.aero.org [130.221.26.26]) by rushe.aero.org (8.11.7p1+Sun/8.11.6) with ESMTP id i43NMOD15326; Mon, 3 May 2004 16:22:24 -0700 (PDT) From: Chris Landauer <cal(a)rushg.aero.org> Received: (from cal@localhost) by calamari.aero.org (8.11.6/8.11.6) id i43NP2U93869; Mon, 3 May 2004 16:25:02 -0700 (PDT) (envelope-from cal) Date: Mon, 3 May 2004 16:25:02 -0700 (PDT) Message-Id: <200405032325.i43NP2U93869(a)calamari.aero.org> To: math-fun(a)mailman.xmission.com X-SA-Exim-Mail-From: cal(a)rushg.aero.org X-SA-Exim-Version: 3.1 (built Mon Jan 26 13:00:24 MST 2004) X-SA-Exim-Scanned: Yes Cc: cal(a)rushg.aero.org Subject: [math-fun] divisor chains are strange X-BeenThere: math-fun(a)mailman.xmission.com X-Mailman-Version: 2.1.4 Precedence: list Reply-To: math-fun <math-fun(a)mailman.xmission.com> List-Id: math-fun <math-fun.mailman.xmission.com> List-Unsubscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=unsubscribe> List-Archive: <http://mailman.xmission.com/cgi-bin/mailman/private/math-fun> List-Post: <mailto:math-fun@mailman.xmission.com> List-Help: <mailto:math-fun-request@mailman.xmission.com?subject=help> List-Subscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=subscribe> Sender: math-fun-bounces+njas=research.att.com(a)mailman.xmission.com Errors-To: math-fun-bounces+njas=research.att.com(a)mailman.xmission.com X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on mail-gray.research.att.com X-Spam-Level: X-Spam-Status: No, hits=-104.9 required=4.0 tests=AWL,BAYES_00, USER_IN_ALL_SPAM_TO autolearn=ham version=2.63 Status: RO R.K.Guy wrote (approximately) a division chain for n is the sequence of numbers from 1 to n, arranged so that each term is a divisor of the sum of the preceding ones i wrote a program to compute them so i could look at them here are the counts up to n=27 (more are pending; this is the simplest program i could think of, and it is not very smart) 1 of length 1 1 of length 2 1 of length 3 1 of length 4 2 of length 5 2 of length 6 3 of length 7 6 of length 8 14 of length 9 12 of length 10 15 of length 11 17 of length 12 29 of length 13 24 of length 14 178 of length 15 128 of length 16 156 of length 17 140 of length 18 182 of length 19 507 of length 20 2210 of length 21 1636 of length 22 2272 of length 23 9455 of length 24 17437 of length 25 30202 of length 26 102292 of length 27 there are two curious phenomena i notice by looking at the chains themselves: there are very limited choices for the first and last element (1) first elements seem to be all large values (mostly, but not always, all consecutive values from some point up to the largest) (2) last elements seem to be extremely limited (mostly, but not always, no more than 3 different choices) i'll put the entire file, with all chains, counts, and the program, out on the web when the program finishes (unless somebody disputes the counts, in which case i'll post only the program on the web instead - it is only 160 lines of C code - and we can figure out what i did wrong) more soon, cal Chris Landauer Aerospace Integration Science Center The Aerospace Corporation cal(a)aero.org _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >From math-fun-bounces+njas=research.att.com(a)mailman.xmission.com Mon May 3 19:30:01 2004 Received: from mail-gray.research.att.com (mail-gray.research.att.com [135.207.30.104]) by fry.research.att.com (SGI-8.9.3p2/8.8.7) with ESMTP id TAA39798 for <njas(a)fry.research.att.com>; 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Mon, 03 May 2004 17:29:27 -0600 Received: from [130.221.88.102] (helo=mhultra.aero.org) by mgr7.xmission.com with esmtp (Exim 4.30) id 1BKmsI-0004dr-Sk for math-fun(a)mailman.xmission.com; Mon, 03 May 2004 17:29:26 -0600 Received: from rushe.aero.org ([130.221.24.10] [130.221.24.10]) by mhultra.aero.org with ESMTP for math-fun(a)mailman.xmission.com; Mon, 3 May 2004 16:29:12 -0700 Received: from calamari.aero.org (calamari.aero.org [130.221.26.26]) by rushe.aero.org (8.11.7p1+Sun/8.11.6) with ESMTP id i43NTBD16369; Mon, 3 May 2004 16:29:11 -0700 (PDT) From: Chris Landauer <cal(a)rushg.aero.org> Received: (from cal@localhost) by calamari.aero.org (8.11.6/8.11.6) id i43NVmG93905; Mon, 3 May 2004 16:31:48 -0700 (PDT) (envelope-from cal) Date: Mon, 3 May 2004 16:31:48 -0700 (PDT) Message-Id: <200405032331.i43NVmG93905(a)calamari.aero.org> To: math-fun(a)mailman.xmission.com X-SA-Exim-Mail-From: cal(a)rushg.aero.org X-SA-Exim-Version: 3.1 (built Mon Jan 26 13:00:24 MST 2004) X-SA-Exim-Scanned: Yes Cc: cal(a)rushg.aero.org Subject: [math-fun] divisor chain last elements - duh X-BeenThere: math-fun(a)mailman.xmission.com X-Mailman-Version: 2.1.4 Precedence: list Reply-To: math-fun <math-fun(a)mailman.xmission.com> List-Id: math-fun <math-fun.mailman.xmission.com> List-Unsubscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=unsubscribe> List-Archive: <http://mailman.xmission.com/cgi-bin/mailman/private/math-fun> List-Post: <mailto:math-fun@mailman.xmission.com> List-Help: <mailto:math-fun-request@mailman.xmission.com?subject=help> List-Subscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=subscribe> Sender: math-fun-bounces+njas=research.att.com(a)mailman.xmission.com Errors-To: math-fun-bounces+njas=research.att.com(a)mailman.xmission.com X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on mail-gray.research.att.com X-Spam-Level: X-Spam-Status: No, hits=-104.9 required=4.0 tests=AWL,BAYES_00, USER_IN_ALL_SPAM_TO autolearn=ham version=2.63 Status: RO hihi, all - i wrote that the last elements of these divisor chains were very limited - well, they have to divide n*(n+1)/2 and be no larger than n, and there are a lot of small primes more later, cal Chris Landauer Aerospace Integration Science Center The Aerospace Corporation cal(a)aero.org _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >From math-fun-bounces+njas=research.att.com(a)mailman.xmission.com Mon May 3 20:36:42 2004 Received: from mail-gray.research.att.com (mail-gray.research.att.com [135.207.30.104]) by fry.research.att.com (SGI-8.9.3p2/8.8.7) with ESMTP id UAA31618 for <njas(a)fry.research.att.com>; Mon, 3 May 2004 20:36:42 -0400 (EDT) Received: by mail-gray.research.att.com (Postfix) id EF31C75807F; Mon, 3 May 2004 20:36:11 -0400 (EDT) Delivered-To: njas(a)research.att.com Received: by mail-gray.research.att.com (Postfix, from userid 612) id E181B758080; Mon, 3 May 2004 20:36:11 -0400 (EDT) Received: from mail-dark.research.att.com (mail-dark.research.att.com [192.20.225.112]) by mail-gray.research.att.com (Postfix) with ESMTP id 00F7A75807F for <njas(a)research.att.com>; Mon, 3 May 2004 20:36:11 -0400 (EDT) Received: from mailman.xmission.com (mailman.xmission.com [198.60.22.29]) by mail-dark.research.att.com (Postfix) with ESMTP id C4F4CC82E9 for <njas(a)research.att.com>; Mon, 3 May 2004 20:36:10 -0400 (EDT) Received: from localhost ([127.0.0.1] helo=mailman.xmission.com) by mailman.xmission.com with esmtp (Exim 3.36 #1 (Debian)) id 1BKnus-0003x9-03 for <njas(a)research.att.com>; Mon, 03 May 2004 18:36:10 -0600 Received: from mgr8.xmission.com ([198.60.22.208] ident=mail) by mailman.xmission.com with esmtp (Exim 3.36 #1 (Debian)) id 1BKnup-0003x2-00 for <math-fun(a)mailman.xmission.com>; Mon, 03 May 2004 18:36:07 -0600 Received: from [130.221.88.102] (helo=mhultra.aero.org) by mgr8.xmission.com with esmtp (Exim 4.30) id 1BKnun-0003cO-RI for math-fun(a)mailman.xmission.com; Mon, 03 May 2004 18:36:05 -0600 Received: from rushe.aero.org ([130.221.24.10] [130.221.24.10]) by mhultra.aero.org with ESMTP for math-fun(a)mailman.xmission.com; Mon, 3 May 2004 17:33:43 -0700 Received: from calamari.aero.org (calamari.aero.org [130.221.26.26]) by rushe.aero.org (8.11.7p1+Sun/8.11.6) with ESMTP id i440XfD25255; Mon, 3 May 2004 17:33:42 -0700 (PDT) From: Chris Landauer <cal(a)rushg.aero.org> Received: (from cal@localhost) by calamari.aero.org (8.11.6/8.11.6) id i440aEJ94328; Mon, 3 May 2004 17:36:14 -0700 (PDT) (envelope-from cal) Date: Mon, 3 May 2004 17:36:14 -0700 (PDT) Message-Id: <200405040036.i440aEJ94328(a)calamari.aero.org> To: math-fun(a)mailman.xmission.com X-SA-Exim-Mail-From: cal(a)rushg.aero.org X-SA-Exim-Version: 3.1 (built Mon Jan 26 13:00:24 MST 2004) X-SA-Exim-Scanned: Yes Cc: cal(a)rushg.aero.org Subject: [math-fun] divisor chains counts fixed X-BeenThere: math-fun(a)mailman.xmission.com X-Mailman-Version: 2.1.4 Precedence: list Reply-To: math-fun <math-fun(a)mailman.xmission.com> List-Id: math-fun <math-fun.mailman.xmission.com> List-Unsubscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=unsubscribe> List-Archive: <http://mailman.xmission.com/cgi-bin/mailman/private/math-fun> List-Post: <mailto:math-fun@mailman.xmission.com> List-Help: <mailto:math-fun-request@mailman.xmission.com?subject=help> List-Subscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=subscribe> Sender: math-fun-bounces+njas=research.att.com(a)mailman.xmission.com Errors-To: math-fun-bounces+njas=research.att.com(a)mailman.xmission.com X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on mail-gray.research.att.com X-Spam-Level: X-Spam-Status: No, hits=-104.9 required=4.0 tests=BAYES_00, USER_IN_ALL_SPAM_TO autolearn=ham version=2.63 Status: RO hihi, all - oops - my program did not count sequences in which one of the terms was equal to the sum of the previous terms (< instead of <= in the loop condition 8-() anyway, here are the revised counts 1 chains of length 1 1 chains of length 2 2 chains of length 3 2 chains of length 4 4 chains of length 5 5 chains of length 6 7 chains of length 7 7 chains of length 8 24 chains of length 9 22 chains of length 10 29 chains of length 11 39 chains of length 12 67 chains of length 13 55 chains of length 14 386 chains of length 15 235 chains of length 16 312 chains of length 17 347 chains of length 18 451 chains of length 19 1319 chains of length 20 5320 chains of length 21 3220 chains of length 22 4489 chains of length 23 20237 chains of length 24 36580 chains of length 25 52875 chains of length 26 197103 chains of length 27 more soon, cal _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >From seqfan-owner(a)ext.jussieu.fr Mon May 3 21:56:29 2004 Received: from mail-brown.research.att.com (mail-brown.research.att.com [135.207.30.105]) by fry.research.att.com (SGI-8.9.3p2/8.8.7) with ESMTP id VAA29398 for <njas(a)fry.research.att.com>; 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Tue, 4 May 2004 03:53:44 +0200 (CEST) Received: (from sympa@localhost) by idf.ext.jussieu.fr (8.12.6/8.12.6/Submit) id i441rJgl072946; Tue, 4 May 2004 03:53:19 +0200 (CEST) X-Authentication-Warning: idf.ext.jussieu.fr: sympa set sender to seqfan-owner(a)ext.jussieu.fr using -f Received: from shiva.jussieu.fr (shiva.jussieu.fr [134.157.0.129]) by idf.ext.jussieu.fr (8.12.6/jtpda-5.4) with ESMTP id i441qsUk072888 for <seqfan(a)ext.jussieu.fr>; Tue, 4 May 2004 03:52:54 +0200 (CEST) Received: from shell.cas.usf.edu (shell.cas.usf.edu [131.247.211.200]) by shiva.jussieu.fr (8.12.11/jtpda-5.4) with ESMTP id i441qDQt099935 for <seqfan(a)ext.jussieu.fr>; Tue, 4 May 2004 03:52:22 +0200 (CEST) X-Ids: 164 Received: from shell.cas.usf.edu (localhost.localdomain [127.0.0.1]) by shell.cas.usf.edu (8.12.10/8.12.10) with ESMTP id i441qD7U017385 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NO); Mon, 3 May 2004 21:52:13 -0400 Received: from localhost (eclark@localhost) by shell.cas.usf.edu (8.12.10/8.12.10/Submit) with ESMTP id i441qCYg017381; Mon, 3 May 2004 21:52:12 -0400 X-Authentication-Warning: shell.cas.usf.edu: eclark owned process doing -bs Date: Mon, 3 May 2004 21:52:12 -0400 (EDT) From: Edwin Clark <eclark(a)math.usf.edu> X-X-Sender: eclark(a)shell.cas.usf.edu To: Richard Guy <rkg(a)cpsc.ucalgary.ca> Cc: Math Fun <math-fun(a)mailman.xmission.com>, seqfan <seqfan(a)ext.jussieu.fr>, "Vaderlind, Paul -- Paul Vaderlind" <paul(a)matematik.su.se>, Paul Vaderlind <paul(a)math.su.se>, Loren Larson <lllarsson(a)earthlink.net> Subject: Re: Divisor chains In-Reply-To: <Pine.LNX.4.44.0405031403560.12751-100000@csl> Message-ID: <Pine.LNX.4.58.0405032040200.17207(a)shell.cas.usf.edu> References: <Pine.LNX.4.44.0405031403560.12751-100000@csl> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Miltered: at shiva.jussieu.fr with ID 4096F78B.000 by Joe's j-chkmail (http://j-chkmail.ensmp.fr) X-Miltered: at shiva.jussieu.fr with ID 4096F74D.001 by Joe's j-chkmail (http://j-chkmail.ensmp.fr) X-Antivirus: scanned by sophie at shiva.jussieu.fr X-Antivirus: scanned by sophie at shiva.jussieu.fr X-Loop: seqfan(a)ext.jussieu.fr X-Sequence: 3951 Errors-To: seqfan-owner(a)ext.jussieu.fr Precedence: list X-no-archive: yes List-Id: <seqfan.ext.jussieu.fr> List-Help: <mailto:sympa@ext.jussieu.fr?subject=help> List-Subscribe: <mailto:sympa@ext.jussieu.fr?subject=subscribe%20seqfan> List-Unsubscribe: <mailto:sympa@ext.jussieu.fr?subject=unsubscribe%20seqfan> List-Post: <mailto:seqfan@ext.jussieu.fr> List-Owner: <mailto:seqfan-request@ext.jussieu.fr> X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on mail-brown.research.att.com X-Spam-Level: X-Spam-Status: No, hits=-104.9 required=4.0 tests=BAYES_00, USER_IN_ALL_SPAM_TO autolearn=ham version=2.63 Status: RO On Mon, 3 May 2004, Richard Guy wrote: > The following might be an acceptable sequence > for Neil Sloane's OEIS, but someone needs to > do some work > > 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ... > 1 1 1 1 1 1 1 5 4 1 2 5 5 4 ... > I compute up to n = 19 the following number of divisor chains: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 1 1 1 1 1 1 5 4 3 2 8 4 6 47 44 6 37 6 which differs from your values at 10, 12, 13, 14. It's interesting that a(15) > a(16). One might think powers of 2 would give larger values. If anyone cares to check my chains from 10 to 16 here they are: 10 2 4 8 6 5 7 3 9 1 10 2 4 8 3 9 6 7 1 5 10 2 6 3 7 4 8 5 9 1 11 1 4 8 6 10 5 9 2 7 3 11 1 2 7 3 8 4 9 5 10 6 12 3 5 10 2 1 11 4 8 7 9 6 12 2 7 1 11 3 9 5 10 4 8 6 12 2 7 3 8 4 9 5 10 6 11 1 12 4 8 3 9 6 2 11 5 10 7 1 12 4 8 6 10 5 9 2 7 3 11 1 12 6 9 3 2 8 4 11 5 10 7 1 12 3 5 10 2 8 4 11 1 7 9 6 12 2 1 5 10 3 11 4 8 7 9 6 13 1 2 8 6 10 4 11 5 3 9 12 7 13 1 2 8 6 10 4 11 5 12 9 3 7 13 1 2 8 12 4 10 5 11 6 9 3 7 13 1 2 8 3 9 4 10 5 11 6 12 7 14 2 8 6 10 4 11 5 3 9 12 7 13 1 14 2 8 6 10 4 11 5 12 9 3 7 13 1 14 2 8 3 9 12 4 13 1 11 7 6 10 5 14 2 8 3 9 4 10 5 11 6 12 7 13 1 14 2 8 12 4 10 5 11 6 9 3 7 13 1 14 2 8 12 9 3 4 13 1 11 7 6 10 5 15 3 2 1 7 14 6 4 13 5 10 8 11 9 12 15 5 10 6 12 8 14 7 11 2 9 1 4 13 3 15 5 4 3 9 12 2 10 6 11 1 13 7 14 8 15 5 4 2 13 1 8 12 3 7 14 6 9 11 10 15 5 10 3 11 4 12 2 1 9 6 13 7 14 8 15 5 10 3 11 4 6 9 7 14 12 8 13 1 2 15 3 6 4 14 7 1 10 5 13 2 8 11 9 12 15 3 2 4 12 9 5 10 6 11 1 13 7 14 8 15 1 2 9 3 10 4 11 5 12 6 13 7 14 8 15 5 10 3 11 1 9 6 12 4 2 13 7 14 8 15 5 4 12 9 3 2 10 6 11 1 13 7 14 8 15 1 4 2 11 3 9 5 10 12 6 13 7 14 8 15 1 8 12 6 2 11 5 10 14 7 13 4 9 3 15 5 10 2 1 11 4 6 9 7 14 12 8 13 3 15 3 6 4 14 7 1 2 13 5 10 8 11 9 12 15 5 10 6 3 13 4 7 9 12 14 1 11 2 8 15 5 10 2 4 9 3 12 6 11 1 13 7 14 8 15 5 4 2 13 3 6 1 7 14 10 8 11 9 12 15 5 10 3 11 1 9 2 4 12 6 13 7 14 8 15 5 10 6 3 13 4 14 7 11 8 12 9 1 2 15 5 4 2 13 1 8 6 9 7 14 12 3 11 10 15 5 10 6 9 3 4 13 1 11 7 12 2 14 8 15 5 10 6 2 1 13 4 14 7 11 8 12 9 3 15 5 1 7 2 6 3 13 4 14 10 8 11 9 12 15 5 10 2 4 12 6 9 3 11 1 13 7 14 8 15 5 4 2 13 3 14 7 1 8 12 6 9 11 10 15 5 10 6 12 4 2 9 3 11 1 13 7 14 8 15 5 4 2 13 3 6 8 7 9 12 14 1 11 10 15 5 10 6 12 8 14 7 11 2 1 13 4 9 3 15 5 10 6 3 13 4 14 7 11 8 2 1 9 12 15 3 9 1 2 10 4 11 5 12 6 13 7 14 8 15 5 10 6 9 3 8 14 7 11 2 1 13 4 12 15 1 2 3 7 14 6 4 13 5 10 8 11 9 12 15 1 8 12 9 5 10 6 11 7 14 2 4 13 3 15 5 10 6 3 13 4 8 2 11 7 14 1 9 12 15 5 10 6 9 3 12 4 2 11 1 13 7 14 8 15 5 10 2 1 11 4 12 3 9 6 13 7 14 8 15 5 2 11 3 6 14 7 9 8 10 1 13 4 12 15 5 10 3 11 4 8 14 2 6 13 7 1 9 12 15 5 10 6 2 1 13 4 14 7 11 8 3 9 12 15 3 9 1 14 6 4 13 5 7 11 8 12 2 10 15 3 6 12 9 5 10 4 2 11 1 13 7 14 8 15 1 8 3 9 6 2 11 5 10 14 7 13 4 12 15 5 4 12 9 3 8 14 2 6 13 7 1 11 10 15 3 2 10 5 7 14 8 1 13 6 4 11 9 12 15 5 10 6 3 13 4 14 7 1 2 8 11 9 12 15 5 4 3 9 12 8 14 2 6 13 7 1 11 10 16 4 10 15 3 2 5 11 6 12 7 13 8 14 9 1 16 2 6 12 9 5 10 15 3 13 7 14 8 1 11 4 16 8 12 9 5 10 6 11 7 14 2 4 13 3 15 1 16 8 12 9 5 10 15 3 13 7 1 11 2 14 6 4 16 8 3 9 6 2 11 5 10 14 7 13 4 12 15 1 16 8 12 9 15 10 5 3 13 7 14 2 6 1 11 4 16 8 12 9 5 10 15 3 13 7 14 2 6 1 11 4 16 2 6 4 14 3 15 10 7 11 8 12 9 13 5 1 16 8 6 15 9 2 14 5 3 13 7 1 11 10 12 4 16 4 2 11 3 9 5 10 12 6 13 7 14 8 15 1 16 2 9 3 10 4 11 5 12 6 13 7 14 8 15 1 16 2 9 3 15 5 10 12 6 13 7 14 8 1 11 4 16 8 12 6 2 11 5 10 14 7 13 4 9 3 15 1 16 8 2 13 3 14 7 9 12 6 15 5 10 1 11 4 16 8 2 13 1 5 15 3 7 14 6 9 11 10 12 4 16 2 6 12 9 15 10 5 3 13 7 14 8 1 11 4 16 8 12 3 13 2 9 7 14 6 15 5 10 1 11 4 16 2 9 3 15 5 10 6 11 1 13 7 14 8 12 4 16 8 2 13 3 6 1 7 14 5 15 9 11 10 12 4 16 8 6 3 11 4 12 5 13 2 10 15 7 14 9 1 16 8 12 3 13 4 14 5 15 9 11 10 6 7 1 2 16 8 12 3 13 4 14 7 11 2 15 5 10 6 9 1 16 8 12 9 15 5 13 6 14 7 3 2 10 1 11 4 16 8 3 9 12 2 10 5 13 6 14 7 15 1 11 4 16 8 3 9 6 2 11 5 10 14 7 13 1 15 12 4 16 4 10 15 3 8 14 7 11 2 6 12 9 13 5 1 16 2 3 7 14 6 4 13 5 10 8 11 9 12 15 1 16 8 4 14 3 15 10 7 11 2 6 12 9 13 5 1 16 2 9 3 10 5 15 6 11 1 13 7 14 8 12 4 16 8 2 13 3 14 7 9 12 6 10 5 15 1 11 4 16 8 12 9 3 2 10 5 13 6 14 7 15 1 11 4 16 8 2 13 1 10 5 11 6 9 3 14 7 15 12 4 16 8 12 3 13 2 9 7 14 6 10 5 15 1 11 4 16 8 12 9 15 10 5 3 13 7 1 11 2 14 6 4 16 4 10 15 9 2 14 7 11 8 12 6 3 13 5 1 16 8 4 14 6 12 10 7 11 2 15 3 9 13 5 1 16 2 9 3 10 8 12 5 13 6 14 7 15 1 11 4 16 4 10 6 12 8 14 7 11 2 15 3 9 13 5 1 16 8 12 3 13 4 14 7 11 2 10 5 15 6 9 1 16 4 10 15 3 6 2 14 7 11 8 12 9 13 5 1 16 2 9 3 10 5 15 12 6 13 7 14 8 1 11 4 16 4 2 11 3 9 15 12 6 13 7 14 8 10 5 1 16 8 12 6 14 4 10 7 11 2 15 3 9 13 5 1 16 8 12 9 15 6 11 7 14 2 4 13 3 10 5 1 >From seqfan-owner(a)ext.jussieu.fr Mon May 3 23:10:30 2004 Received: from mail-gray.research.att.com (mail-gray.research.att.com [135.207.30.104]) by fry.research.att.com (SGI-8.9.3p2/8.8.7) with ESMTP id XAA13187 for <njas(a)fry.research.att.com>; Mon, 3 May 2004 23:10:30 -0400 (EDT) Received: by mail-gray.research.att.com (Postfix) id 5A4D475801C; Mon, 3 May 2004 23:10:00 -0400 (EDT) Delivered-To: njas(a)research.att.com Received: by mail-gray.research.att.com (Postfix, from userid 612) id 50FBC75801E; Mon, 3 May 2004 23:10:00 -0400 (EDT) Received: from mail-dark.research.att.com (mail-dark.research.att.com [192.20.225.112]) by mail-gray.research.att.com (Postfix) with ESMTP id 6E74375801C for <njas(a)research.att.com>; Mon, 3 May 2004 23:09:58 -0400 (EDT) Received: from shiva.jussieu.fr (shiva.jussieu.fr [134.157.0.129]) by mail-dark.research.att.com (Postfix) with ESMTP id A3D7EC8313 for <njas(a)research.att.com>; Mon, 3 May 2004 23:09:56 -0400 (EDT) Received: from idf.ext.jussieu.fr (idf.ext.jussieu.fr [134.157.81.129]) by shiva.jussieu.fr (8.12.11/jtpda-5.4) with ESMTP id i4437U1K070085 ; Tue, 4 May 2004 05:07:30 +0200 (CEST) Received: from idf.ext.jussieu.fr (localhost [127.0.0.1]) by idf.ext.jussieu.fr (8.12.6/jtpda-5.4) with ESMTP id i4437xUk077815 ; Tue, 4 May 2004 05:07:59 +0200 (CEST) Received: (from sympa@localhost) by idf.ext.jussieu.fr (8.12.6/8.12.6/Submit) id i4437xXh077811; Tue, 4 May 2004 05:07:59 +0200 (CEST) X-Authentication-Warning: idf.ext.jussieu.fr: sympa set sender to seqfan-owner(a)ext.jussieu.fr using -f Received: from shiva.jussieu.fr (shiva.jussieu.fr [134.157.0.129]) by idf.ext.jussieu.fr (8.12.6/jtpda-5.4) with ESMTP id i4437gUk077783 for <seqfan(a)ext.jussieu.fr>; Tue, 4 May 2004 05:07:42 +0200 (CEST) Received: from mtiwmhc11.worldnet.att.net (mtiwmhc11.worldnet.att.net [204.127.131.115]) by shiva.jussieu.fr (8.12.11/jtpda-5.4) with ESMTP id i44373bB011194 for <seqfan(a)ext.jussieu.fr>; Tue, 4 May 2004 05:07:09 +0200 (CEST) X-Ids: 164 Received: from plastereddragon (159.cambridge-10rh15rt.ma.dial-access.att.net[12.76.185.159]) by worldnet.att.net (mtiwmhc11) with SMTP id <2004050403073111100ifdj6e>; Tue, 4 May 2004 03:07:32 +0000 Message-ID: <000c01c43185$d159a120$0200a8c0@plastereddragon> Reply-To: "Chuck Seggelin" <seqfan(a)plastereddragon.com> From: "Chuck Seggelin" <seqfan(a)plastereddragon.com> To: "Richard Guy" <rkg(a)cpsc.ucalgary.ca>, "Math Fun" <math-fun(a)mailman.xmission.com>, <seqfan(a)ext.jussieu.fr> Cc: "Vaderlind, Paul -- Paul Vaderlind" <paul(a)matematik.su.se>, "Paul Vaderlind" <paul(a)math.su.se>, "Loren Larson" <lllarsson(a)earthlink.net> References: <Pine.LNX.4.44.0405031403560.12751-100000@csl> Subject: Re: [seqfan] Divisor chains Date: Mon, 3 May 2004 23:13:44 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2800.1106 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2800.1106 X-Miltered: at shiva.jussieu.fr with ID 409708F2.000 by Joe's j-chkmail (http://j-chkmail.ensmp.fr) X-Miltered: at shiva.jussieu.fr with ID 409708D7.000 by Joe's j-chkmail (http://j-chkmail.ensmp.fr) X-Antivirus: scanned by sophie at shiva.jussieu.fr X-Antivirus: scanned by sophie at shiva.jussieu.fr X-Loop: seqfan(a)ext.jussieu.fr X-Sequence: 3952 Errors-To: seqfan-owner(a)ext.jussieu.fr Precedence: list X-no-archive: yes List-Id: <seqfan.ext.jussieu.fr> List-Help: <mailto:sympa@ext.jussieu.fr?subject=help> List-Subscribe: <mailto:sympa@ext.jussieu.fr?subject=subscribe%20seqfan> List-Unsubscribe: <mailto:sympa@ext.jussieu.fr?subject=unsubscribe%20seqfan> List-Post: <mailto:seqfan@ext.jussieu.fr> List-Owner: <mailto:seqfan-request@ext.jussieu.fr> X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on mail-gray.research.att.com X-Spam-Level: X-Spam-Status: No, hits=-103.7 required=4.0 tests=AWL,BAYES_00, USER_IN_ALL_SPAM_TO autolearn=ham version=2.63 Status: RO Wrote a litte recursive Maple code to solve this problem. My output agrees with Richards for n from 1 to 9 and 11. For n=10 I find 3 chains (Richard's + 2), for n=12 I find 8 chains (Richard's + 3), for n=13 I find 4 (Richard's 4'th n=13 chain includes "3" twice), for n=14 I find 6 chains (Richard's + 2). Other than the errant chain for 13, the chains Richard provides appear to be correct. This makes the sequence up to 14 look like this: 1, 1, 1, 1, 1, 1, 1, 5, 4, 3, 2, 8, 4, 6 >From there on the values start getting much larger (here is n=15 to n=24): 47, 44, 6, 37, 6, 166, 462, 232, 372, 2130 If anyone is interested I'm happy to share the code. I'm waiting on n=25 right now. The trend is generally upward so I suspect there are MANY chains for n=25. Given how many chains there are for these numbers, I would expect the number of chains for n=37 to be pretty high, although for some primes (such as 17 and 19) there are small numbers of chains. Richard do you suggest n=37 because you believe there to be very few chains? -- Chuck ----- Original Message ----- From: "Richard Guy" <rkg(a)cpsc.ucalgary.ca> To: "Math Fun" <math-fun(a)mailman.xmission.com>; <seqfan(a)ext.jussieu.fr> Cc: "Vaderlind, Paul -- Paul Vaderlind" <paul(a)matematik.su.se>; "Paul Vaderlind" <paul(a)math.su.se>; "Loren Larson" <lllarsson(a)earthlink.net> Sent: Monday, May 03, 2004 4:20 PM Subject: [seqfan] Divisor chains > The following might be an acceptable sequence > for Neil Sloane's OEIS, but someone needs to > do some work > > 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ... > 1 1 1 1 1 1 1 5 4 1 2 5 5 4 ... > > It arises from a problem I got recently from > Paul Vaderlind. > > If the sequence of numbers from 1 to 37, > is arranged so that each term is a divisor > of the sum of preceding ones, starting > 37, 1, ... > what is the next term? > > The answer is either 2 or 19. P'r'aps > I won't spoil your fun by pointing out > which. But I will spoil it by asking: > How do you know that there is such a > `divisor chain' ? > > The sequence is (my present state of knowledge > of) the number of divisor chains of length n. > Here are the ones I found > > 1 2 1 3 1 2 4 2 3 1 > > 5 1 2 4 3 6 2 4 3 5 1 7 1 2 5 3 6 4 > > 8 2 5 3 6 4 7 1 > 8 4 2 7 3 1 5 6 > 8 4 2 7 3 6 5 1 > 8 4 3 5 1 7 2 6 > 8 4 6 3 7 2 5 1 > > 9 1 2 4 8 6 5 7 3 > 9 1 2 6 3 7 4 8 5 > 9 3 4 8 6 5 7 2 1 > 9 3 6 2 1 7 4 8 5 > > 10 2 4 8 3 9 6 7 1 5 > > 11 1 2 7 3 8 4 9 5 10 6 > 11 1 4 8 6 10 5 9 2 7 3 > > 12 2 1 5 10 3 11 4 8 7 9 6 > 12 2 7 3 8 4 9 5 10 6 11 1 > 12 3 5 10 2 8 4 11 1 7 9 6 > 12 4 8 3 9 6 2 11 5 10 7 1 > 12 4 8 6 10 5 9 2 7 3 11 1 > > 13 1 2 8 3 9 4 10 5 11 6 12 7 > 13 1 2 8 6 10 4 11 5 12 9 3 7 > 13 1 2 8 6 10 4 11 5 3 9 12 7 > 13 1 2 8 12 4 10 5 11 3 9 3 7 > 13 1 2 8 12 4 10 5 11 6 9 3 7 > > 14 2 8 6 10 4 11 5 3 9 12 7 13 1 > 14 2 8 6 10 4 11 5 12 9 3 7 13 1 > 14 2 8 12 4 10 5 11 6 9 3 7 13 1 > 14 2 8 12 9 3 4 13 1 11 7 6 10 5 > > Enough sins of omission, to say nothing of > commission, for today. Please check and extend. > Any ideas for proving anything? R. > > >From seqfan-owner(a)ext.jussieu.fr Mon May 3 23:21:18 2004 Received: from mail-brown.research.att.com (mail-brown.research.att.com [135.207.30.105]) by fry.research.att.com (SGI-8.9.3p2/8.8.7) with ESMTP id XAA99766 for <njas(a)fry.research.att.com>; Mon, 3 May 2004 23:21:18 -0400 (EDT) Received: by mail-brown.research.att.com (Postfix) id 536C92D4086; Mon, 3 May 2004 23:20:48 -0400 (EDT) Delivered-To: njas(a)research.att.com Received: by mail-brown.research.att.com (Postfix, from userid 612) id 471372D40A9; Mon, 3 May 2004 23:20:48 -0400 (EDT) Received: from mail-dark.research.att.com (mail-dark.research.att.com [192.20.225.112]) by mail-brown.research.att.com (Postfix) with ESMTP id 8378F2D4086 for <njas(a)research.att.com>; Mon, 3 May 2004 23:20:46 -0400 (EDT) Received: from shiva.jussieu.fr (shiva.jussieu.fr [134.157.0.129]) by mail-dark.research.att.com (Postfix) with ESMTP id D7DAFC8452 for <njas(a)research.att.com>; Mon, 3 May 2004 23:20:44 -0400 (EDT) Received: from idf.ext.jussieu.fr (idf.ext.jussieu.fr [134.157.81.129]) by shiva.jussieu.fr (8.12.11/jtpda-5.4) with ESMTP id i443IUUX071753 ; Tue, 4 May 2004 05:18:30 +0200 (CEST) Received: from idf.ext.jussieu.fr (localhost [127.0.0.1]) by idf.ext.jussieu.fr (8.12.6/jtpda-5.4) with ESMTP id i443IxUk078690 ; Tue, 4 May 2004 05:18:59 +0200 (CEST) Received: (from sympa@localhost) by idf.ext.jussieu.fr (8.12.6/8.12.6/Submit) id i443IxWo078687; Tue, 4 May 2004 05:18:59 +0200 (CEST) X-Authentication-Warning: idf.ext.jussieu.fr: sympa set sender to seqfan-owner(a)ext.jussieu.fr using -f Received: from shiva.jussieu.fr (shiva.jussieu.fr [134.157.0.129]) by idf.ext.jussieu.fr (8.12.6/jtpda-5.4) with ESMTP id i443IrUk078654 for <seqfan(a)ext.jussieu.fr>; Tue, 4 May 2004 05:18:54 +0200 (CEST) Received: from mtiwmhc12.worldnet.att.net (mtiwmhc12.worldnet.att.net [204.127.131.116]) by shiva.jussieu.fr (8.12.11/jtpda-5.4) with ESMTP id i443IHk5071579 for <seqfan(a)ext.jussieu.fr>; Tue, 4 May 2004 05:18:23 +0200 (CEST) X-Ids: 166 Received: from plastereddragon (159.cambridge-10rh15rt.ma.dial-access.att.net[12.76.185.159]) by worldnet.att.net (mtiwmhc12) with SMTP id <2004050403184411200mgb7je>; Tue, 4 May 2004 03:18:44 +0000 Message-ID: <001401c43187$6273d1c0$0200a8c0@plastereddragon> Reply-To: "Chuck Seggelin" <seqfan(a)plastereddragon.com> From: "Chuck Seggelin" <seqfan(a)plastereddragon.com> To: "Richard Guy" <rkg(a)cpsc.ucalgary.ca>, "Math Fun" <math-fun(a)mailman.xmission.com>, <seqfan(a)ext.jussieu.fr> Cc: "Vaderlind, Paul -- Paul Vaderlind" <paul(a)matematik.su.se>, "Paul Vaderlind" <paul(a)math.su.se>, "Loren Larson" <lllarsson(a)earthlink.net> References: <Pine.LNX.4.44.0405031403560.12751-100000@csl> <000c01c43185$d159a120$0200a8c0@plastereddragon> Subject: Re: [seqfan] Divisor chains Date: Mon, 3 May 2004 23:25:00 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2800.1106 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2800.1106 X-Miltered: at shiva.jussieu.fr with ID 40970B86.003 by Joe's j-chkmail (http://j-chkmail.ensmp.fr) X-Miltered: at shiva.jussieu.fr with ID 40970B79.000 by Joe's j-chkmail (http://j-chkmail.ensmp.fr) X-Antivirus: scanned by sophie at shiva.jussieu.fr X-Antivirus: scanned by sophie at shiva.jussieu.fr X-Loop: seqfan(a)ext.jussieu.fr X-Sequence: 3953 Errors-To: seqfan-owner(a)ext.jussieu.fr Precedence: list X-no-archive: yes List-Id: <seqfan.ext.jussieu.fr> List-Help: <mailto:sympa@ext.jussieu.fr?subject=help> List-Subscribe: <mailto:sympa@ext.jussieu.fr?subject=subscribe%20seqfan> List-Unsubscribe: <mailto:sympa@ext.jussieu.fr?subject=unsubscribe%20seqfan> List-Post: <mailto:seqfan@ext.jussieu.fr> List-Owner: <mailto:seqfan-request@ext.jussieu.fr> X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on mail-brown.research.att.com X-Spam-Level: X-Spam-Status: No, hits=-104.9 required=4.0 tests=BAYES_00, USER_IN_ALL_SPAM_TO autolearn=ham version=2.63 Status: RO Sorry for the double-post... for n=25 there are 1589 chains. First 25 terms are therefore: 1, 1, 1, 1, 1, 1, 1, 5, 4, 3, 2, 8, 4, 6, 47, 44, 6, 37, 6, 166, 462, 232, 372, 2130, 1589 -- Chuck ----- Original Message ----- From: "Chuck Seggelin" <seqfan(a)plastereddragon.com> To: "Richard Guy" <rkg(a)cpsc.ucalgary.ca>; "Math Fun" <math-fun(a)mailman.xmission.com>; <seqfan(a)ext.jussieu.fr> Cc: "Vaderlind, Paul -- Paul Vaderlind" <paul(a)matematik.su.se>; "Paul Vaderlind" <paul(a)math.su.se>; "Loren Larson" <lllarsson(a)earthlink.net> Sent: Monday, May 03, 2004 11:13 PM Subject: Re: [seqfan] Divisor chains > Wrote a litte recursive Maple code to solve this problem. My output agrees > with Richards for n from 1 to 9 and 11. For n=10 I find 3 chains (Richard's > + 2), for n=12 I find 8 chains (Richard's + 3), for n=13 I find 4 (Richard's > 4'th n=13 chain includes "3" twice), for n=14 I find 6 chains (Richard's + > 2). Other than the errant chain for 13, the chains Richard provides appear > to be correct. > > This makes the sequence up to 14 look like this: > > 1, 1, 1, 1, 1, 1, 1, 5, 4, 3, 2, 8, 4, 6 > > From there on the values start getting much larger (here is n=15 to n=24): > > 47, 44, 6, 37, 6, 166, 462, 232, 372, 2130 > > If anyone is interested I'm happy to share the code. I'm waiting on n=25 > right now. The trend is generally upward so I suspect there are MANY chains > for n=25. > > Given how many chains there are for these numbers, I would expect the number > of chains for n=37 to be pretty high, although for some primes (such as 17 > and 19) there are small numbers of chains. Richard do you suggest n=37 > because you believe there to be very few chains? > > -- Chuck > > ----- Original Message ----- > From: "Richard Guy" <rkg(a)cpsc.ucalgary.ca> > To: "Math Fun" <math-fun(a)mailman.xmission.com>; <seqfan(a)ext.jussieu.fr> > Cc: "Vaderlind, Paul -- Paul Vaderlind" <paul(a)matematik.su.se>; "Paul > Vaderlind" <paul(a)math.su.se>; "Loren Larson" <lllarsson(a)earthlink.net> > Sent: Monday, May 03, 2004 4:20 PM > Subject: [seqfan] Divisor chains > > > > The following might be an acceptable sequence > > for Neil Sloane's OEIS, but someone needs to > > do some work > > > > 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ... > > 1 1 1 1 1 1 1 5 4 1 2 5 5 4 ... > > > > It arises from a problem I got recently from > > Paul Vaderlind. > > > > If the sequence of numbers from 1 to 37, > > is arranged so that each term is a divisor > > of the sum of preceding ones, starting > > 37, 1, ... > > what is the next term? > > > > The answer is either 2 or 19. P'r'aps > > I won't spoil your fun by pointing out > > which. But I will spoil it by asking: > > How do you know that there is such a > > `divisor chain' ? > > > > The sequence is (my present state of knowledge > > of) the number of divisor chains of length n. > > Here are the ones I found > > > > 1 2 1 3 1 2 4 2 3 1 > > > > 5 1 2 4 3 6 2 4 3 5 1 7 1 2 5 3 6 4 > > > > 8 2 5 3 6 4 7 1 > > 8 4 2 7 3 1 5 6 > > 8 4 2 7 3 6 5 1 > > 8 4 3 5 1 7 2 6 > > 8 4 6 3 7 2 5 1 > > > > 9 1 2 4 8 6 5 7 3 > > 9 1 2 6 3 7 4 8 5 > > 9 3 4 8 6 5 7 2 1 > > 9 3 6 2 1 7 4 8 5 > > > > 10 2 4 8 3 9 6 7 1 5 > > > > 11 1 2 7 3 8 4 9 5 10 6 > > 11 1 4 8 6 10 5 9 2 7 3 > > > > 12 2 1 5 10 3 11 4 8 7 9 6 > > 12 2 7 3 8 4 9 5 10 6 11 1 > > 12 3 5 10 2 8 4 11 1 7 9 6 > > 12 4 8 3 9 6 2 11 5 10 7 1 > > 12 4 8 6 10 5 9 2 7 3 11 1 > > > > 13 1 2 8 3 9 4 10 5 11 6 12 7 > > 13 1 2 8 6 10 4 11 5 12 9 3 7 > > 13 1 2 8 6 10 4 11 5 3 9 12 7 > > 13 1 2 8 12 4 10 5 11 3 9 3 7 > > 13 1 2 8 12 4 10 5 11 6 9 3 7 > > > > 14 2 8 6 10 4 11 5 3 9 12 7 13 1 > > 14 2 8 6 10 4 11 5 12 9 3 7 13 1 > > 14 2 8 12 4 10 5 11 6 9 3 7 13 1 > > 14 2 8 12 9 3 4 13 1 11 7 6 10 5 > > > > Enough sins of omission, to say nothing of > > commission, for today. Please check and extend. > > Any ideas for proving anything? R. > > > > > > >From math-fun-bounces+njas=research.att.com(a)mailman.xmission.com Tue May 4 00:47:38 2004 Received: from mail-gray.research.att.com (mail-gray.research.att.com [135.207.30.104]) by fry.research.att.com (SGI-8.9.3p2/8.8.7) with ESMTP id AAA74402 for <njas(a)fry.research.att.com>; Tue, 4 May 2004 00:47:38 -0400 (EDT) Received: by mail-gray.research.att.com (Postfix) id 72790758071; Tue, 4 May 2004 00:47:08 -0400 (EDT) Delivered-To: njas(a)research.att.com Received: by mail-gray.research.att.com (Postfix, from userid 612) id 64BE3758082; Tue, 4 May 2004 00:47:08 -0400 (EDT) Received: from mail-dark.research.att.com (mail-dark.research.att.com [192.20.225.112]) by mail-gray.research.att.com (Postfix) with ESMTP id 2ABF4758071 for <njas(a)research.att.com>; Tue, 4 May 2004 00:47:07 -0400 (EDT) Received: from mailman.xmission.com (mailman.xmission.com [198.60.22.29]) by mail-dark.research.att.com (Postfix) with ESMTP id F1082C84B9 for <njas(a)research.att.com>; Tue, 4 May 2004 00:47:06 -0400 (EDT) Received: from localhost ([127.0.0.1] helo=mailman.xmission.com) by mailman.xmission.com with esmtp (Exim 3.36 #1 (Debian)) id 1BKrpi-00046F-06 for <njas(a)research.att.com>; Mon, 03 May 2004 22:47:06 -0600 Received: from mgr8.xmission.com ([198.60.22.208] ident=mail) by mailman.xmission.com with esmtp (Exim 3.36 #1 (Debian)) id 1BKrpf-00045q-00 for <math-fun(a)mailman.xmission.com>; Mon, 03 May 2004 22:47:03 -0600 Received: from [130.221.88.102] (helo=mhultra.aero.org) by mgr8.xmission.com with esmtp (Exim 4.30) id 1BKrpd-0005fO-Sh for math-fun(a)mailman.xmission.com; Mon, 03 May 2004 22:47:01 -0600 Received: from rushe.aero.org ([130.221.24.10] [130.221.24.10]) by mhultra.aero.org with ESMTP for math-fun(a)mailman.xmission.com; Mon, 3 May 2004 21:46:32 -0700 Received: from calamari.aero.org (calamari.aero.org [130.221.26.26]) by rushe.aero.org (8.11.7p1+Sun/8.11.6) with ESMTP id i444kVD26687; Mon, 3 May 2004 21:46:31 -0700 (PDT) From: Chris Landauer <cal(a)rushg.aero.org> Received: (from cal@localhost) by calamari.aero.org (8.11.6/8.11.6) id i444n9795633; Mon, 3 May 2004 21:49:09 -0700 (PDT) (envelope-from cal) Date: Mon, 3 May 2004 21:49:09 -0700 (PDT) Message-Id: <200405040449.i444n9795633(a)calamari.aero.org> To: math-fun(a)mailman.xmission.com X-SA-Exim-Mail-From: cal(a)rushg.aero.org X-SA-Exim-Version: 3.1 (built Mon Jan 26 13:00:24 MST 2004) X-SA-Exim-Scanned: Yes Cc: cal(a)rushg.aero.org Subject: [math-fun] counts for related divisor chains X-BeenThere: math-fun(a)mailman.xmission.com X-Mailman-Version: 2.1.4 Precedence: list Reply-To: math-fun <math-fun(a)mailman.xmission.com> List-Id: math-fun <math-fun.mailman.xmission.com> List-Unsubscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=unsubscribe> List-Archive: <http://mailman.xmission.com/cgi-bin/mailman/private/math-fun> List-Post: <mailto:math-fun@mailman.xmission.com> List-Help: <mailto:math-fun-request@mailman.xmission.com?subject=help> List-Subscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=subscribe> Sender: math-fun-bounces+njas=research.att.com(a)mailman.xmission.com Errors-To: math-fun-bounces+njas=research.att.com(a)mailman.xmission.com X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on mail-gray.research.att.com X-Spam-Level: X-Spam-Status: No, hits=-104.9 required=4.0 tests=BAYES_00, USER_IN_ALL_SPAM_TO autolearn=ham version=2.63 Status: R hihi, all - several people have reminded me that there are some sequences related to the divisor chains one of them must start with n one of them must start with a divisor of n*(n+1)/2 (so it can form a cycle) actually, if the requirement is that each cyclic permutation of the sequence is also a divisor chain, then there are none (except the trivial one 1 for n=1), since not all integers start divisor chains for any given n>1 all of these subsequent counts were done by manual editing of the sequence output from the first program, so typos are quie possible the first counts are when the sequence must start with n (i call these anchored): 1 chains of length 1 1 chains of length 2 @ 1 chains of length 3 * 1 chains of length 4 @ 1 chains of length 5 * 1 chains of length 6 @ 1 chains of length 7 * 5 chains of length 8 @ 4 chains of length 9 * 3 chains of length 10 @ 2 chains of length 11 8 chains of length 12 4 chains of length 13 * 6 chains of length 14 47 chains of length 15 44 chains of length 16 @ 6 chains of length 17 * 37 chains of length 18 @ 6 chains of length 19 166 chains of length 20 462 chains of length 21 232 chains of length 22 @ 372 chains of length 23 2130 chains of length 24 1589 chains of length 25 * 9093 chains of length 26 @ (* = n and (n+1)/2 are prime powers, n odd) (@ = n/2 and n+1 are prime powers, n even) the second case is when the first element must divide n*(n+1)/2 (i call these cyclic): 1 chains of length 1 1 chains of length 2 2 chains of length 3 1 chains of length 4 2 chains of length 5 0 chains of length 6 3 chains of length 7 0 chains of length 8 5 chains of length 9 0 chains of length 10 6 chains of length 11 0 chains of length 12 6 chains of length 13 0 chains of length 14 145 chains of length 15 0 chains of length 16 22 chains of length 17 2 chains of length 18 27 chains of length 19 165 chains of length 20 57 chains of length 21 0 chains of length 22 516 chains of length 23 2021 chains of length 24 1912 chains of length 25 453 chains of length 26 the third case is when both conditions must hold: 1 chains of length 1 1 chains of length 2 1 chains of length 3 1 chains of length 4 1 chains of length 5 0 chains of length 6 1 chains of length 7 0 chains of length 8 4 chains of length 9 0 chains of length 10 2 chains of length 11 0 chains of length 12 4 chains of length 13 0 chains of length 14 47 chains of length 15 0 chains of length 16 6 chains of length 17 0 chains of length 18 6 chains of length 19 0 chains of length 20 0 chains of length 21 0 chains of length 22 372 chains of length 23 0 chains of length 24 1589 chains of length 25 0 chains of length 26 more later, cal Chris Landauer Aerospace Integration Science Center The Aerospace Corporation cal(a)aero.org _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >From seqfan-owner(a)ext.jussieu.fr Tue May 4 03:32:32 2004 Received: from mail-brown.research.att.com (mail-brown.research.att.com [135.207.30.105]) by fry.research.att.com (SGI-8.9.3p2/8.8.7) with ESMTP id DAA52697 for <njas(a)fry.research.att.com>; Tue, 4 May 2004 03:32:32 -0400 (EDT) Received: by mail-brown.research.att.com (Postfix) id 75C7D2D40BE; Tue, 4 May 2004 03:32:02 -0400 (EDT) Delivered-To: njas(a)research.att.com Received: by mail-brown.research.att.com (Postfix, from userid 612) id 6975D2D40C6; Tue, 4 May 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seqfan-owner(a)ext.jussieu.fr using -f Received: from shiva.jussieu.fr (shiva.jussieu.fr [134.157.0.129]) by idf.ext.jussieu.fr (8.12.6/jtpda-5.4) with ESMTP id i447TqUk098810 for <seqfan(a)ext.jussieu.fr>; Tue, 4 May 2004 09:29:52 +0200 (CEST) Received: from goose.mail.pas.earthlink.net (goose.mail.pas.earthlink.net [207.217.120.18]) by shiva.jussieu.fr (8.12.11/jtpda-5.4) with ESMTP id i447Te1Z023110 for <seqfan(a)ext.jussieu.fr>; Tue, 4 May 2004 09:29:46 +0200 (CEST) X-Ids: 165 Received: from pool0061.cvx22-bradley.dialup.earthlink.net ([209.179.198.61] helo=[209.179.198.66]) by goose.mail.pas.earthlink.net with esmtp (Exim 3.33 #1) id 1BKuN6-0002Kq-00 for seqfan(a)ext.jussieu.fr; Tue, 04 May 2004 00:29:44 -0700 Mime-Version: 1.0 X-Sender: dwasserm(a)pop.earthlink.net Message-Id: <v04210102bcbcf5ae537d(a)[209.179.198.66]> In-Reply-To: <Pine.LNX.4.44.0405031403560.12751-100000@csl> References: <Pine.LNX.4.44.0405031403560.12751-100000@csl> X-mailer: Eudora Pro 4.2.1 Macintosh Date: Tue, 4 May 2004 00:30:16 -0700 To: seqfan(a)ext.jussieu.fr From: David Wasserman <dwasserm(a)earthlink.com> Subject: Re: Divisor chains Content-Type: text/plain; charset="us-ascii" X-j-chkmail-Score: MSGID : 40974664.000 on shiva.jussieu.fr : j-chkmail score : X : 0/50 1 X-Miltered: at shiva.jussieu.fr with ID 4097464B.001 by Joe's j-chkmail (http://j-chkmail.ensmp.fr) X-Miltered: at shiva.jussieu.fr with ID 40974664.000 by Joe's j-chkmail (http://j-chkmail.ensmp.fr) X-Antivirus: scanned by sophie at shiva.jussieu.fr X-Antivirus: scanned by sophie at shiva.jussieu.fr X-Loop: seqfan(a)ext.jussieu.fr X-Sequence: 3957 Errors-To: seqfan-owner(a)ext.jussieu.fr Precedence: list X-no-archive: yes List-Id: <seqfan.ext.jussieu.fr> List-Help: <mailto:sympa@ext.jussieu.fr?subject=help> List-Subscribe: <mailto:sympa@ext.jussieu.fr?subject=subscribe%20seqfan> List-Unsubscribe: <mailto:sympa@ext.jussieu.fr?subject=unsubscribe%20seqfan> List-Post: <mailto:seqfan@ext.jussieu.fr> List-Owner: <mailto:seqfan-request@ext.jussieu.fr> X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on mail-brown.research.att.com X-Spam-Level: X-Spam-Status: No, hits=-92.3 required=4.0 tests=AWL,BAYES_00, USER_IN_ALL_SPAM_TO autolearn=ham version=2.63 Status: R >From the results that have been posted, I expect that there are many divisor chains for n = 37, but no one has exhibited any. So here's two of them: 37 1 2 4 11 5 3 7 14 6 15 35 28 8 22 33 21 36 18 34 20 12 31 13 32 16 29 17 30 10 25 23 26 24 27 9 19 37 1 2 4 11 5 3 7 14 6 15 35 28 8 22 33 21 36 18 34 20 12 31 13 32 16 29 17 30 10 25 23 26 24 27 9 19 I found these in about 20 minutes of computer time. >From math-fun-bounces+njas=research.att.com(a)mailman.xmission.com Tue May 4 03:48:25 2004 Received: from mail-gray.research.att.com (mail-gray.research.att.com [135.207.30.104]) by fry.research.att.com (SGI-8.9.3p2/8.8.7) with ESMTP id DAA30738 for <njas(a)fry.research.att.com>; Tue, 4 May 2004 03:48:25 -0400 (EDT) Received: by mail-gray.research.att.com (Postfix) id 37AF7758085; Tue, 4 May 2004 03:47:55 -0400 (EDT) Delivered-To: njas(a)research.att.com Received: by mail-gray.research.att.com (Postfix, from userid 612) id 2873E758086; Tue, 4 May 2004 03:47:55 -0400 (EDT) Received: from mail-dark.research.att.com (mail-dark.research.att.com [192.20.225.112]) by mail-gray.research.att.com (Postfix) with ESMTP id 02D6D758085 for <njas(a)research.att.com>; Tue, 4 May 2004 03:47:54 -0400 (EDT) Received: from mailman.xmission.com (mailman.xmission.com [198.60.22.29]) by mail-dark.research.att.com (Postfix) with ESMTP id CB7B7C865E for <njas(a)research.att.com>; Tue, 4 May 2004 03:47:53 -0400 (EDT) Received: from localhost ([127.0.0.1] helo=mailman.xmission.com) by mailman.xmission.com with esmtp (Exim 3.36 #1 (Debian)) id 1BKuef-0006pL-01 for <njas(a)research.att.com>; Tue, 04 May 2004 01:47:53 -0600 Received: from mgr8.xmission.com ([198.60.22.208] ident=mail) by mailman.xmission.com with esmtp (Exim 3.36 #1 (Debian)) id 1BKueb-0006pE-00 for <math-fun(a)mailman.xmission.com>; Tue, 04 May 2004 01:47:49 -0600 Received: from [207.217.120.50] (helo=avocet.mail.pas.earthlink.net) by mgr8.xmission.com with esmtp (Exim 4.30) id 1BKueZ-0003ap-Rm for math-fun(a)mailman.xmission.com; Tue, 04 May 2004 01:47:47 -0600 Received: from daisy.psp.pas.earthlink.net ([207.217.78.223]) by avocet.mail.pas.earthlink.net with esmtp (Exim 3.33 #1) id 1BKueY-00017U-00; Tue, 04 May 2004 00:47:46 -0700 Message-ID: <13375876.1083656866205.JavaMail.root(a)daisy.psp.pas.earthlink.net> Date: Tue, 4 May 2004 00:47:45 -0700 (GMT-07:00) From: Daniel Asimov <dasimov(a)earthlink.net> To: math-fun <math-fun(a)mailman.xmission.com>, seqfan(a)ext.jussieu.fr Mime-Version: 1.0 X-Mailer: Earthlink Zoo Mail 1.0 Subject: Re: [math-fun] Divisor chains Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-SA-Exim-Mail-From: dasimov(a)earthlink.net X-SA-Exim-Version: 3.1 (built Mon Jan 26 13:00:24 MST 2004) X-SA-Exim-Scanned: Yes Cc: Paul Vaderlind <paul(a)math.su.se> X-BeenThere: math-fun(a)mailman.xmission.com X-Mailman-Version: 2.1.4 Precedence: list Reply-To: Daniel Asimov <dasimov(a)earthlink.net>, math-fun <math-fun(a)mailman.xmission.com> List-Id: math-fun <math-fun.mailman.xmission.com> List-Unsubscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=unsubscribe> List-Archive: <http://mailman.xmission.com/cgi-bin/mailman/private/math-fun> List-Post: <mailto:math-fun@mailman.xmission.com> List-Help: <mailto:math-fun-request@mailman.xmission.com?subject=help> List-Subscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=subscribe> Sender: math-fun-bounces+njas=research.att.com(a)mailman.xmission.com Errors-To: math-fun-bounces+njas=research.att.com(a)mailman.xmission.com X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on mail-gray.research.att.com X-Spam-Level: X-Spam-Status: No, hits=-104.9 required=4.0 tests=BAYES_00, USER_IN_ALL_SPAM_TO autolearn=ham version=2.63 Status: R RIchard Guy wrote: << How do you know that there is such a `divisor chain' ? It took quite a while for me to realize that for {1,2,...,n}, if we let M = floor(n/2), the following permutations always work: For n even: M+1, 1, M+2, 2, ..., n-1, M-1, n, M. For n odd: M+1, 1, M+2, 2, ..., M-1, n-1, M, n The proof is both magical and completely obvious. E.g., for n = 10: 6, 1, 7, 2, 8, 3, 9, 4, 10, 5; for n = 37: 19, 1, 20, 2, ..., 35, 17, 36, 18, 37. Still don't know how to show there exists a 37-chain beginning with 37, 1. But since the final term must divide 37*19, the third term must be 2 in any such chain. D'uh! --Dan _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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Re: [math-fun] Divisor chains
by Daniel Asimov 04 May '04

04 May '04

RIchard Guy wrote: << How do you know that there is such a `divisor chain' ? It took quite a while for me to realize that for {1,2,...,n}, if we let M = floor(n/2), the following permutations always work: For n even: M+1, 1, M+2, 2, ..., n-1, M-1, n, M. For n odd: M+1, 1, M+2, 2, ..., M-1, n-1, M, n The proof is both magical and completely obvious. E.g., for n = 10: 6, 1, 7, 2, 8, 3, 9, 4, 10, 5; for n = 37: 19, 1, 20, 2, ..., 35, 17, 36, 18, 37. Still don't know how to show there exists a 37-chain beginning with 37, 1. But since the final term must divide 37*19, the third term must be 2 in any such chain. D'uh! --Dan

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[math-fun] penultimate divisor counts output
by Chris Landauer 04 May '04

04 May '04

hihi, all - yes, well, there were some typos - now the program counts them, so i don't have to there will be a few more numbers tomorrow (hopefully, n=37 will appear), but for now, here is what i have 1 chains of length 1, 1 anchored, 1 cyclic, 1 both 1 chains of length 2, 1 anchored, 0 cyclic, 0 both 2 chains of length 3, 1 anchored, 2 cyclic, 1 both 2 chains of length 4, 1 anchored, 0 cyclic, 0 both 4 chains of length 5, 1 anchored, 2 cyclic, 1 both 5 chains of length 6, 1 anchored, 0 cyclic, 0 both 7 chains of length 7, 1 anchored, 3 cyclic, 1 both 7 chains of length 8, 5 anchored, 0 cyclic, 0 both 24 chains of length 9, 4 anchored, 5 cyclic, 4 both 22 chains of length 10, 3 anchored, 0 cyclic, 0 both 29 chains of length 11, 2 anchored, 6 cyclic, 2 both 39 chains of length 12, 8 anchored, 0 cyclic, 0 both 67 chains of length 13, 4 anchored, 6 cyclic, 4 both 55 chains of length 14, 6 anchored, 0 cyclic, 0 both 386 chains of length 15, 47 anchored, 147 cyclic, 47 both 235 chains of length 16, 44 anchored, 1 cyclic, 0 both 312 chains of length 17, 6 anchored, 22 cyclic, 6 both 347 chains of length 18, 37 anchored, 2 cyclic, 0 both 451 chains of length 19, 6 anchored, 27 cyclic, 6 both 1319 chains of length 20, 166 anchored, 165 cyclic, 0 both 5320 chains of length 21, 462 anchored, 519 cyclic, 462 both 3220 chains of length 22, 232 anchored, 0 cyclic, 0 both 4489 chains of length 23, 372 anchored, 516 cyclic, 372 both 20237 chains of length 24, 2130 anchored, 2021 cyclic, 0 both 36580 chains of length 25, 1589 anchored, 1912 cyclic, 1589 both 52875 chains of length 26, 9093 anchored, 506 cyclic, 0 both 197103 chains of length 27, 20896 anchored, 45658 cyclic, 20896 both 216562 chains of length 28, 20314 anchored, 514 cyclic, 0 both (for a sequence to be both cyclic and anchored, the first element must both divide n*(n+1)/2 and be equal to n, so n must be odd) one more tomorrow, cal Chris Landauer Aerospace Integration Science Center The Aerospace Corporation cal(a)aero.org

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[math-fun] counts for related divisor chains
by Chris Landauer 03 May '04

03 May '04

hihi, all - several people have reminded me that there are some sequences related to the divisor chains one of them must start with n one of them must start with a divisor of n*(n+1)/2 (so it can form a cycle) actually, if the requirement is that each cyclic permutation of the sequence is also a divisor chain, then there are none (except the trivial one 1 for n=1), since not all integers start divisor chains for any given n>1 all of these subsequent counts were done by manual editing of the sequence output from the first program, so typos are quie possible the first counts are when the sequence must start with n (i call these anchored): 1 chains of length 1 1 chains of length 2 @ 1 chains of length 3 * 1 chains of length 4 @ 1 chains of length 5 * 1 chains of length 6 @ 1 chains of length 7 * 5 chains of length 8 @ 4 chains of length 9 * 3 chains of length 10 @ 2 chains of length 11 8 chains of length 12 4 chains of length 13 * 6 chains of length 14 47 chains of length 15 44 chains of length 16 @ 6 chains of length 17 * 37 chains of length 18 @ 6 chains of length 19 166 chains of length 20 462 chains of length 21 232 chains of length 22 @ 372 chains of length 23 2130 chains of length 24 1589 chains of length 25 * 9093 chains of length 26 @ (* = n and (n+1)/2 are prime powers, n odd) (@ = n/2 and n+1 are prime powers, n even) the second case is when the first element must divide n*(n+1)/2 (i call these cyclic): 1 chains of length 1 1 chains of length 2 2 chains of length 3 1 chains of length 4 2 chains of length 5 0 chains of length 6 3 chains of length 7 0 chains of length 8 5 chains of length 9 0 chains of length 10 6 chains of length 11 0 chains of length 12 6 chains of length 13 0 chains of length 14 145 chains of length 15 0 chains of length 16 22 chains of length 17 2 chains of length 18 27 chains of length 19 165 chains of length 20 57 chains of length 21 0 chains of length 22 516 chains of length 23 2021 chains of length 24 1912 chains of length 25 453 chains of length 26 the third case is when both conditions must hold: 1 chains of length 1 1 chains of length 2 1 chains of length 3 1 chains of length 4 1 chains of length 5 0 chains of length 6 1 chains of length 7 0 chains of length 8 4 chains of length 9 0 chains of length 10 2 chains of length 11 0 chains of length 12 4 chains of length 13 0 chains of length 14 47 chains of length 15 0 chains of length 16 6 chains of length 17 0 chains of length 18 6 chains of length 19 0 chains of length 20 0 chains of length 21 0 chains of length 22 372 chains of length 23 0 chains of length 24 1589 chains of length 25 0 chains of length 26 more later, cal Chris Landauer Aerospace Integration Science Center The Aerospace Corporation cal(a)aero.org

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[math-fun] Divisor chains
by Richard Guy 03 May '04

03 May '04

The following might be an acceptable sequence for Neil Sloane's OEIS, but someone needs to do some work 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ... 1 1 1 1 1 1 1 5 4 1 2 5 5 4 ... It arises from a problem I got recently from Paul Vaderlind. If the sequence of numbers from 1 to 37, is arranged so that each term is a divisor of the sum of preceding ones, starting 37, 1, ... what is the next term? The answer is either 2 or 19. P'r'aps I won't spoil your fun by pointing out which. But I will spoil it by asking: How do you know that there is such a `divisor chain' ? The sequence is (my present state of knowledge of) the number of divisor chains of length n. Here are the ones I found 1 2 1 3 1 2 4 2 3 1 5 1 2 4 3 6 2 4 3 5 1 7 1 2 5 3 6 4 8 2 5 3 6 4 7 1 8 4 2 7 3 1 5 6 8 4 2 7 3 6 5 1 8 4 3 5 1 7 2 6 8 4 6 3 7 2 5 1 9 1 2 4 8 6 5 7 3 9 1 2 6 3 7 4 8 5 9 3 4 8 6 5 7 2 1 9 3 6 2 1 7 4 8 5 10 2 4 8 3 9 6 7 1 5 11 1 2 7 3 8 4 9 5 10 6 11 1 4 8 6 10 5 9 2 7 3 12 2 1 5 10 3 11 4 8 7 9 6 12 2 7 3 8 4 9 5 10 6 11 1 12 3 5 10 2 8 4 11 1 7 9 6 12 4 8 3 9 6 2 11 5 10 7 1 12 4 8 6 10 5 9 2 7 3 11 1 13 1 2 8 3 9 4 10 5 11 6 12 7 13 1 2 8 6 10 4 11 5 12 9 3 7 13 1 2 8 6 10 4 11 5 3 9 12 7 13 1 2 8 12 4 10 5 11 3 9 3 7 13 1 2 8 12 4 10 5 11 6 9 3 7 14 2 8 6 10 4 11 5 3 9 12 7 13 1 14 2 8 6 10 4 11 5 12 9 3 7 13 1 14 2 8 12 4 10 5 11 6 9 3 7 13 1 14 2 8 12 9 3 4 13 1 11 7 6 10 5 Enough sins of omission, to say nothing of commission, for today. Please check and extend. Any ideas for proving anything? R.

4 6

[math-fun] divisor chains counts fixed
by Chris Landauer 03 May '04

03 May '04

hihi, all - oops - my program did not count sequences in which one of the terms was equal to the sum of the previous terms (< instead of <= in the loop condition 8-() anyway, here are the revised counts 1 chains of length 1 1 chains of length 2 2 chains of length 3 2 chains of length 4 4 chains of length 5 5 chains of length 6 7 chains of length 7 7 chains of length 8 24 chains of length 9 22 chains of length 10 29 chains of length 11 39 chains of length 12 67 chains of length 13 55 chains of length 14 386 chains of length 15 235 chains of length 16 312 chains of length 17 347 chains of length 18 451 chains of length 19 1319 chains of length 20 5320 chains of length 21 3220 chains of length 22 4489 chains of length 23 20237 chains of length 24 36580 chains of length 25 52875 chains of length 26 197103 chains of length 27 more soon, cal

2 1

[math-fun] divisor chain last elements - duh
by Chris Landauer 03 May '04

03 May '04

hihi, all - i wrote that the last elements of these divisor chains were very limited - well, they have to divide n*(n+1)/2 and be no larger than n, and there are a lot of small primes more later, cal Chris Landauer Aerospace Integration Science Center The Aerospace Corporation cal(a)aero.org

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[math-fun] divisor chains are strange
by Chris Landauer 03 May '04

03 May '04

R.K.Guy wrote (approximately) a division chain for n is the sequence of numbers from 1 to n, arranged so that each term is a divisor of the sum of the preceding ones i wrote a program to compute them so i could look at them here are the counts up to n=27 (more are pending; this is the simplest program i could think of, and it is not very smart) 1 of length 1 1 of length 2 1 of length 3 1 of length 4 2 of length 5 2 of length 6 3 of length 7 6 of length 8 14 of length 9 12 of length 10 15 of length 11 17 of length 12 29 of length 13 24 of length 14 178 of length 15 128 of length 16 156 of length 17 140 of length 18 182 of length 19 507 of length 20 2210 of length 21 1636 of length 22 2272 of length 23 9455 of length 24 17437 of length 25 30202 of length 26 102292 of length 27 there are two curious phenomena i notice by looking at the chains themselves: there are very limited choices for the first and last element (1) first elements seem to be all large values (mostly, but not always, all consecutive values from some point up to the largest) (2) last elements seem to be extremely limited (mostly, but not always, no more than 3 different choices) i'll put the entire file, with all chains, counts, and the program, out on the web when the program finishes (unless somebody disputes the counts, in which case i'll post only the program on the web instead - it is only 160 lines of C code - and we can figure out what i did wrong) more soon, cal Chris Landauer Aerospace Integration Science Center The Aerospace Corporation cal(a)aero.org

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[math-fun] Erdos number =5 sold for over $1000
by Thane Plambeck 03 May '04

03 May '04

http://cgi.ebay.com/ws/eBayISAPI.dll?ViewItem&item=3189039958 Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.plambeck.org

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[math-fun] Experimental math book, vol 2
by David H Bailey 01 May '04

01 May '04

I am pleased to announce that the second volume of our work experimental mathematics is now out, available from the publisher or from Amazon.com. The second volume gives numerous case studies of experimental mathematics in action, ranging from sequences, series, products, integrals, Fourier series, zeta functions, partitions, primes and polynomials. Some advanced numerical techniques are also presented. The reference info is: Authors: Jonathan M. Borwein, David H. Bailey and Roland Girgensohn Title: "Experimentation in Mathematics: Computational Paths to Discovery" Publisher: A. K. Peters (http://akpeters.com) 2004 If you would like a better taste of what these books are like, a 75-page condensed version is available FREE at: http://www.expmath.info David H Bailey dhbailey(a)lbl.gov

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