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Gilbreath transform of primes p(2k-1); see Comments.
3

%I #12 Sep 25 2023 19:24:14

%S 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 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 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1

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

%C 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 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 Conjectured examples of Gilbreath transforms:

%C If S = A000040 (primes), then G(S) = A000012 = (1,1,1,...)

%C If S = A000045 (Fibonacci numbers), then G(S) = A011655 = (0,1,1,0,1,1,...)

%C If S = A000032 (Lucas number)s, G(S) = (2,1,1,0,1,1,0,1,1,...)

%C If S = A031368 (odd-indexed primes), then G(S) = A358691 = (3,3,3,3,1,1,1,...)

%C If S = A031369, then G(S) = A358692 = (1,3,1,1,1,1,...)

%C Two further conjectured examples:

%C (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 (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 <a href="/index/Ge#Gilbreath">Index entries for sequences related to Gilbreath conjecture and transform</a>

%e Corner of successive absolute difference array (including initial row of primes p(2k-1)):

%e 2 5 11 17 23 31 41 47 59 67

%e 3 6 6 6 8 10 6 12 8 6

%e 3 0 0 2 2 4 6 4 2 4

%e 3 0 2 0 2 2 2 2 2 0

%e 3 2 2 2 0 0 0 0 2 4

%e 1 0 0 2 0 0 0 2 2 0

%e 1 0 2 2 0 0 2 0 2 0

%t z = 130; g[t_] := Abs[Differences[t]]

%t t = Prime[-1 + 2 Range[140]]

%t s[1] = g[t]; s[n_] := g[s[n - 1]];

%t Table[s[n], {n, 1, z}] ;

%t Table[First[s[n]], {n, 1, z}]

%Y Cf. A000040, A031368, A036262, A358692.

%K nonn

%O 1,1

%A _Clark Kimberling_, Nov 27 2022