[seqfan] Re: A generalized Gilbreath's conjecture, or "lizard's effect" for primes

Maximilian Hasler maximilian.hasler at gmail.com
Sat Jun 2 20:00:26 CEST 2012

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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