A031369 -id:A031369 - cp's OEIS frontend

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

Clark Kimberling, Nov 27 2022

nonn

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)).

			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

Index entries for sequences related to Gilbreath conjecture and transform

Cf. A000040, A031368, A036262, A358692.

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}]

A112303 Permutation of primes generated by 3-rowed array shown below.

Original entry on oeis.org

2, 7, 3, 17, 11, 5, 29, 19, 13, 41, 31, 23, 53, 43, 37, 67, 59, 47, 79, 71, 61, 97, 83, 73, 107, 101, 89, 127, 109, 103, 139, 139, 131, 113, 157, 149, 137, 173, 163, 151, 191, 179, 167, 193, 181, 227, 211, 197, 239, 229, 223, 257, 263, 251

Offset: 1

Giovanni Teofilatto, Oct 17 2006, Nov 08 2006

7 17 29 41 53 67 79...(A031377)
11 19 31 43 59 71 83...(A031369)
13 23 37 47 61 73 89...(A031336)

Cf. A115302.

a(n) = A000040(a(p+3q) = a(p)+3q) with p and q positive integers.

a(n) = PrimePi(a(p+3q) = a(p)+3q) with p and q positive integers.

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

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