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A358691 Gilbreath transform of primes p(2k-1); see Comments.

%I A358691 #12 Sep 25 2023 19:24:14
%S A358691 3,3,3,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T A358691 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A358691 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A358691 Gilbreath transform of primes p(2k-1); see Comments.
%C A358691 Suppose that S = (s(k)), for k >= 1, is a sequence of real numbers.  For n >= 1, let g(1,n) = |s(n+1)-s(n)| and g(k,n) = |g(k-1, n+1) - g(k-1,n)| for k >= 2.
%C A358691 We call (g(k,n)) the Gilbreath array of S and (g(n,1)) the Gilbreath transform of S, written as G(S).  If S is the sequences of primes, then the Gilbreath conjecture holds that G(S) consists exclusively of 1's.  It appears that there are many S such that G(S) is eventually periodic.
%C A358691 Conjectured examples of Gilbreath transforms:
%C A358691 If S = A000040 (primes), then G(S) = A000012 = (1,1,1,...)
%C A358691 If S = A000045 (Fibonacci numbers), then G(S) = A011655 = (0,1,1,0,1,1,...)
%C A358691 If S = A000032 (Lucas number)s, G(S) = (2,1,1,0,1,1,0,1,1,...)
%C A358691 If S = A031368 (odd-indexed primes), then G(S) = A358691 = (3,3,3,3,1,1,1,...)
%C A358691 If S = A031369, then G(S) = A358692 = (1,3,1,1,1,1,...)
%C A358691 Two further conjectured examples:
%C A358691 (1) If S is the sequence of primes of the form k*n+2, where k is an odd positive integer and n>=0, then G(S) = (k,k,k,...).
%C A358691 (2) Suppose that (b(n)) is an increasing arithmetic sequence of positive integers r(s) and S is the sequence of primes p(b(n)).  If b(1) = 1, so that S begins with 2, then G(S) is eventually (1,1,1,...); the same holds if b(1) > 1 and S consists of 2 followed by the terms of p(b(n)).
%H A358691 <a href="/index/Ge#Gilbreath">Index entries for sequences related to Gilbreath conjecture and transform</a>
%e A358691 Corner of successive absolute difference array (including initial row of primes p(2k-1)):
%e A358691   2   5  11  17  23   31  41  47  59  67
%e A358691   3   6   6   6   8   10   6  12   8   6
%e A358691   3   0   0   2   2   4    6   4   2   4
%e A358691   3   0   2   0   2   2    2   2   2   0
%e A358691   3   2   2   2   0   0    0   0   2   4
%e A358691   1   0   0   2   0   0    0   2   2   0
%e A358691   1   0   2   2   0   0    2   0   2   0
%t A358691 z = 130; g[t_] := Abs[Differences[t]]
%t A358691 t = Prime[-1 + 2 Range[140]]
%t A358691 s[1] = g[t]; s[n_] := g[s[n - 1]];
%t A358691 Table[s[n], {n, 1, z}] ;
%t A358691 Table[First[s[n]], {n, 1, z}]
%Y A358691 Cf. A000040, A031368, A036262, A358692.
%K A358691 nonn
%O A358691 1,1
%A A358691 _Clark Kimberling_, Nov 27 2022