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

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, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Clark Kimberling, Nov 27 2022

Keywords

• nonn

Comments

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.

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.

Conjectured examples of Gilbreath transforms:

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

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

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

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

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

Two further conjectured examples:

(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,...).

(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)).

Examples

			Corner of successive absolute difference array (including initial row of primes p(2k-1)):
  2   5  11  17  23   31  41  47  59  67
  3   6   6   6   8   10   6  12   8   6
  3   0   0   2   2   4    6   4   2   4
  3   0   2   0   2   2    2   2   2   0
  3   2   2   2   0   0    0   0   2   4
  1   0   0   2   0   0    0   2   2   0
  1   0   2   2   0   0    2   0   2   0
		

Links

• Index entries for sequences related to Gilbreath conjecture and transform

Crossrefs

Cf. A000040, A031368, A036262, A358692.

Programs

Mathematica

z = 130; g[t_] := Abs[Differences[t]]
t = Prime[-1 + 2 Range[140]]
s[1] = g[t]; s[n_] := g[s[n - 1]];
Table[s[n], {n, 1, z}] ;
Table[First[s[n]], {n, 1, z}]