I don't. I see this distribution of a sum of a series of Poisson-like distributions. In the ramps, the first number is always 0. The second number follows the distribution of the length of the ramps, and is Poisson-like (and small). The third number in the ramp follows (very roughly) the distribution of (0,n) pairs, where n is the second number in the ramp. And so on and so forth. The distributions are sufficiently disjoint where the second order effects stay somewhat small, and the distributions are clearly separable visually. The means of the distributions appear to increase roughly exponentially. There are second-order effects, but I think this is the dominant effect and explains the undulations in the overall distribution. It would be interesting to model this as, or fit this to, a random process with much less state. This won't give particular discrete results but it may give some insight into the behavior. Towards this end it may be useful to mark the peaks of the distributions and how those peaks migrate positivewards. -tom On Sun, Jun 16, 2019 at 1:18 PM Cris Moore <moore@santafe.edu> wrote:
interesting! so the distribution is nonuniform, even among small numbers? what’s a good numerical criterion to see if this persists? do we believe that asymptotically the distribution of numbers becomes stable?
C
On Jun 16, 2019, at 11:46 AM, Tomas Rokicki <rokicki@gmail.com> wrote:
Also, at this point I've shown if there's an integer missing from Van Eck's sequence, it is at least 519,068,589.
And 6's are increasingly rare, more so than other small integers (up to about 250B entries). Initially 3's are rare, but this characteristic moves to 4's, then 5's, and now 6's. One could expect this to continue . . .
On Sun, Jun 16, 2019 at 10:39 AM Tomas Rokicki <rokicki@gmail.com> wrote:
Following along Brad's comments, here are the first occurrences of pairs starting with 0 in Van Eck's sequence. Anyone want to try to fill in the value for (0,32) (or (0,n) for n>34)?
Pair 0 0 at 1 Pair 0 1 at 2 Pair 0 2 at 4 Pair 0 3 at 21 Pair 0 4 at 25 Pair 0 5 at 11 Pair 0 6 at 31 Pair 0 7 at 277 Pair 0 8 at 389 Pair 0 9 at 82 Pair 0 10 at 226 Pair 0 11 at 727 Pair 0 12 at 2,936 Pair 0 13 at 1,409 Pair 0 14 at 7,719 Pair 0 15 at 5,625 Pair 0 16 at 5,681 Pair 0 17 at 85,999 Pair 0 18 at 26,707 Pair 0 19 at 546,291 Pair 0 20 at 1,112,930 Pair 0 21 at 702,576 Pair 0 22 at 3,425,418 Pair 0 23 at 10,537,361 Pair 0 24 at 21,301,907 Pair 0 25 at 217,230,901 Pair 0 26 at 108,698,092 Pair 0 27 at 32,381,775 Pair 0 28 at 846,522,987 Pair 0 29 at 851,764,847 Pair 0 30 at 11,692,311,326 Pair 0 31 at 46,163,898,988 Pair 0 32 at ??? Pair 0 33 at 118,456,929,919 Pair 0 34 at 250,327,022,558 Pair 0 35 at ???
On Wed, Jun 12, 2019 at 12:25 AM Brad Klee <bradklee@gmail.com> wrote:
Ouch... this one looks like it could lead to some serious brain damage.
If A171868 lists all integers, then so too does A181391. Due to the rule, A171868 has much slower growth and, as far as I can tell, could even be bounded above. Does A171868 always continue to grow?
I don't know.
--Brad
On Tue, Jun 11, 2019 at 2:48 AM Bill Gosper <billgosper@gmail.com> wrote:
Neil Sloane has Numberphiled a perplexing integer sequence: https://www.youtube.com/watch?v=etMJxB-igrc —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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