[seqfan] Re: A generalized Gilbreath's conjecture, or "lizard's effect" for primes
shevelev at bgu.ac.il
Sat Jun 2 22:53:33 CEST 2012
Thanks, Maximilian, for your comment. In my experiment, I wanted to understand should we, with respect to Gilbreath's conjecture, consider the sequence of primes as a whole, or we can consider it with respect to all primes, beginning with p_N for arbitrary large N? Or, the same, is the behavior of the arbitrary far differences is the same as the first ones? For me the absent of an aftereffect is not so trivial. The terms of my sequence (A212990) depend on sum of many different mutually influencing factors and give a numerical characterization of the considered iterative process which sometimes quite unexpectedly changes depending on the initial p_N. Only in such a sense I said about a "generalization" of Gilbreath's conjecture. It is better to say that I consider a (new?) interesting for me experiment concerning it.
----- Original Message -----
From: Maximilian Hasler <maximilian.hasler at gmail.com>
Date: Saturday, June 2, 2012 21:01
Subject: [seqfan] Re: A generalized Gilbreath's conjecture, or "lizard's effect" for primes
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> On Fri, Jun 1, 2012 at 10:38 AM, Neil Sloane
> <njasloane at gmail.com> wrote:
> > Re generalizations of Gilbreath's conjecture: see references
> in A036262,
> > especially the Odlyzko paper.
> The first (more precisely: uppermost) row of A036262 is sometimes
> referred to as row 0, and/or as its 0-th row.
> (In A036277, "First row" meant the row 1,2,2,... which is not
> the uppermost row.
> I tentatively changed this to "Row 1" and added a comment, not
> yet published.)
> Shouldn't this motivate to put the offset = 0 ?
> (It would be less confusing to refer to A036261 which is the same
> without this "row 0",
> i.e. starting with the first absolute differences in the
> uppermost row.
> However, in A036261, the sequence A036262 is
> declared to be the main entry.)
> Also, both sequences A036261 and A036262 are "diagonals read upwards",
> while the usual convention is different : as a consequence, the
> tabl link shows
> these "rows" in columns when formatted as square array.
> Would this also merit a comment ?
> PS: I think Vladimir's "generalization" is a direct
> consequence of
> the original conjecture,
> since the rows in his calculations are essentially
> [i.e. up to the initial term and a finite number of missing terms]
> the same than in the original calculation,
> and the first column is always decreasing
> [the contrary would mean a term in second position more than
> twice as
> large as the term in the first position, which seems clearly excluded
> by the growth of A036277],
> thus it will necessarily end up at 1, at which point (after a
> total of
> at least n iterations)
> one goes on with a row identical to one of the original calculation.
> Vladimir Shevelev <shevelev at bgu.ac.il>wrote:
> >> Dear SeqFans,
> >> A very known Gilbreath's conjecture states that the k-th
> iteration (k>=1)
> >> of the absolute values of differences of consecutive primes
> always begins
> >> from 1. I believe that, if to consider 2 followed by the
> consecutive primes
> >> beginning with the n-th prime p_n, n>=2, then there exists an
> iteration>> which begins from 1 and, moreover, after the first
> such iteration all other
> >> iterations begin with 1. I call this effect, when the "tail"
> of 1's
> >> appears after a time, "lizard's effect" for primes.
> >> Denote by a(n) (n>=2) the number of the first iteration
> beginning from 1.
> >> Then I obtained by handy a(2)=1, a(3)=2 (for primes
> 2,5,7,11,...), a(4)=2
> >> (for primes 2,7,11,13,...), a(5)=9, a(6)=7, a(7)=14,
> a(8)=10, a(9)=11.
> >> What is the continuation of this sequence?
> >> Regards,
> >> Vladimir
> >> Shevelev Vladimir
> >> _______________________________________________
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> > --
> > Dear Friends, I have now retired from AT&T. New coordinates:
> > Neil J. A. Sloane, President, OEIS Foundation
> > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA
> > Phone: 732 828 6098; home page: http://NeilSloane.com
> > Email: njasloane at gmail.com
> > _______________________________________________
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