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[math-fun] Prime signature functions
by David Wilson 08 May '05

08 May '05

I notice that for some sequences, a(n) is a function of the prime signature of a n, that is, a function of the multiset of exponents in the prime factorization of n. For such a function, a(p) will be the same for all primes p, a(pq) will be the same for any two distinct primes p, a(p^2) will be the same for any prime p, etc. For the moment I will call there prime signature functions. Is there a standard name for such functions? There are prime signature functions which are not multiplicative and vice versa. Examples of prime signature functions that are not multiplicative are A001221 (number of prime divisors of n) and A001222 (number of prime divisors with multiplicity). Examples of multiplicative functions that are not prime signature functions are A000010 (totient) and A000203 (sum of divisors). A function that is both prime signature and multiplicative is A000005 (number of divisors of n). A function is both prime signature and mutliplicative if f(p^e) is strictly a function of e. What made me think of this is that I was recently working on A000028. Membership of n in A000028 depends entirely on the prime signature of n, so its membership function is a prime signature function. - David W. Wilson "Truth is just truth -- You can't have opinions about the truth." - Peter Schickele, from P.D.Q. Bach's oratorio "The Seasonings"

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Re: [math-fun] better posed problem? (small surprise)
by Phil Carmody 07 May '05

07 May '05

From: "Meeussen Wouter (bkarnd)" <wouter.meeussen(a)vandemoortele.com> > > it is a small surprise that, for w = Exp[ 2 Pi I /30 ] alias 1^(1/30) , > w+w^7+w^8+w^18+w^19+w^25 = 0 > no cancellation of symmetric pairs, triples or quintuples in sight, > but w+w^7+w^19+w^25 = -w^13 (* four out of five *) > and w^8+w^18 = -w^28 = +w^13 (* two out of three *) > > could be called "cancellation by missing parts" ;-)) As (a) {0,6,12,18,24} cancels, (b) {0,10,20} cancels, and (c) {0,15} cancels. {5,15,25} cancels, so {5,6,12,18,24,25} cancels. The largest gap (mod 30) between terms is that between 25 and 5, so {0,1,7,13,19,20} is the symmetry-less set with minimal largest member that cancels. You've exploited the gap of 6 between 18 and 24 instead. 2*p*q is easily so minimised - any takers for zeta_210? Phil When inserting a CD, hold down shift to stop the AutoRun feature In the Device Manager, disable the SbcpHid device. http://www.cs.princeton.edu/~jhalderm/cd3/ __________________________________ Do you Yahoo!? Yahoo! Small Business - Try our new resources site! http://smallbusiness.yahoo.com/resources/

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Re: [math-fun] Mark of the Beast reduced to 616
by Henry Baker 06 May '05

06 May '05

Ha, every computer scientist knows that the Mark of the Beast is the Goedel number of EVAL... Maybe Revelations was the first to Goedelize? At 11:37 AM 5/5/2005, Kerry Mitchell wrote: >http://www.canada.com/national/nationalpost/news/toronto/story.html?id=702d… > >Insert your own joke here. > >Kerry

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[math-fun] Mark of the Beast reduced to 616
by lkmitch＠att.net 05 May '05

05 May '05

http://www.canada.com/national/nationalpost/news/toronto/story.html?id=702d… Insert your own joke here. Kerry

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[math-fun] Found in the IHP cellar! 1/2
by Christian Boyer 05 May '05

05 May '05

---- DIOPHANTE magazine I have the chance to have discovered very recently, in the cellar of the Institut Henri Poincaré, Paris, the originals of all the issues of an old magazine on number theory named "Diophante" that everybody thought lost, written in 1948-1952 by André Gérardin at the end of his life. Gérardin (1879-1953), and its first magazine "Sphinx-Oedipe", is one of the most quoted authors referenced by Leonard E. Dickson in his "History of the Theory of Numbers", both Vol 1 and Vol 2. And also one of the most quoted authors referenced by Hugh C. Williams in his 1998 book "Edouard Lucas and Primality Testing". For example, in Diophante, there are several articles written by Aimé Ferrier, holder of the biggest known prime number in 1951: his prime is still today the largest prime found without electronic computer. http://mathworld.wolfram.com/FerriersPrime.html The IHP library will reprint this magazine for some other French libraries. If you know other libraries, out of France, which may be interested by this reprint, send me a message at cboyer(a)club-internet.fr. I will pass the info to the IHP library. To interested people, I can also send my preface published in this reprint (PDF file... written in French). Best regards. Christian.

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Re: [math-fun] Greedy Change
by Jeffrey Shallit 04 May '05

04 May '05

Yes, there is an algorithm. See http://citeseer.ist.psu.edu/pearson94polynomialtime.html Jeffrey Shallit > > I understand that with US coins of denominations (100, 50, 25, 10, 5, > > 1) cents, making change by the greedy algorithm (at each step select > > the largest possible coin) leads to optimal change (fewest coins) for > > all amounts. > > > > There are other conceivable sets of denominations, e.,g, (10, 6, 1), > > where the greedy algorithm fails (for example, the greedy algorithm > > gives 18 = 10+6+1+1 whereas 18 = 6+6+6 uses fewer coins). > > > > Is there an algorithm for determining if a given set of coins gives > > optimal change via the greedy algorithm? > > > > - David W. Wilson

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[math-fun] Greedy Change
by David Wilson 04 May '05

04 May '05

I understand that with US coins of denominations (100, 50, 25, 10, 5, 1) cents, making change by the greedy algorithm (at each step select the largest possible coin) leads to optimal change (fewest coins) for all amounts. There are other conceivable sets of denominations, e.,g, (10, 6, 1), where the greedy algorithm fails (for example, the greedy algorithm gives 18 = 10+6+1+1 whereas 18 = 6+6+6 uses fewer coins). Is there an algorithm for determining if a given set of coins gives optimal change via the greedy algorithm? - David W. Wilson "Truth is just truth -- You can't have opinions about the truth." - Peter Schickele, from P.D.Q. Bach's oratorio "The Seasonings"

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RE: [math-fun] elliptic mean
by Daniel Asimov 02 May '05

02 May '05

<< Must a Mean associate? -- Rich Well, from ((x+y)/2 + z)/2 = x/4 + y/4/ + z/2 we see that even the arithmetic mean fails to associate. On the other hand, I have in the past worked with mean associates (though that was not obligatory). --Dan

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RE: [math-fun] ~s solid angle <- trihedral angles
by Daniel Asimov 02 May '05

02 May '05

Hi, Bill. I'm always interested in your many identities (no, not aliases), but with my e-mail set to my favorite font (Garamond), they come across mangled somewhat. In trying what used to always work -- just copying your text to a primitive editor (Notepad) to an equal-width font, the text no longer reconstructs itself. Even after pushing some obvious things into place, I'm still not sure what you wrote. E.g., your trihedral post today came out after massaging as follows in Courier New: (cos(c) + cos(b) + cos(a) + 1) 2 acos(-------------------------------------- - 1). (cos(a) + 1) (cos(b) + 1) (cos(c) + 1) Duh, cos(c) + cos(b) + cos(a) + 1 2 acos(----------------------------). a b c 4 cos(-) cos(-) cos(-) 2 2 2 Any suggestions? Regards, Dan

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[math-fun] ~s solid angle <- trihedral angles
by R. William Gosper 02 May '05

02 May '05

A couple of years ago I here gave 2 (cos(c) + cos(b) + cos(a) + 1) acos(-------------------------------------- - 1). (cos(a) + 1) (cos(b) + 1) (cos(c) + 1) Duh, cos(c) + cos(b) + cos(a) + 1 2 acos(----------------------------). a b c 4 cos(-) cos(-) cos(-) 2 2 2 Imaginary result means you violated the triangle inequality. Tsk. --rwg PS, for those interested in trig exercises, this started out as cos(a) - cos(b) cos(c) cos(b) - cos(a) cos(c) acos(----------------------) + acos(----------------------) sin(b) sin(c) sin(a) sin(c) cos(c) - cos(a) cos(b) + acos(----------------------) - %pi sin(a) sin(b) For those with an unhealthy interest in trig exercises, www.ippi.com/rwg/trigbelt.dvi

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