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

1,1

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)).

Links

Table of n, a(n) for n=1..86.

Index entries for sequences related to Gilbreath conjecture and transform

Example

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

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}]

Crossrefs

Cf. A000040, A031368, A036262, A358692.

Sequence in context: A107760 A180560 A320085 * A172368 A138070 A081334

Adjacent sequences: A358688 A358689 A358690 * A358692 A358693 A358694

Keyword

nonn

Author

Clark Kimberling, Nov 27 2022

Status

approved