

A358691


Gilbreath transform of primes p(2k1); see Comments.


3



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
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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(k1, n+1)  g(k1,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 = 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 (oddindexed primes), then G(S) = A358691 = (3,3,3,3,1,1,1,...)
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



EXAMPLE

Corner of successive absolute difference array (including initial row of primes p(2k1)):
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



KEYWORD

nonn


AUTHOR



STATUS

approved



