A358692 Gilbreath transform of primes p(2k) with 2 prefixed; see Comments.

1, 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, 1, 1

Offset

1,2

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)f| 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. See A358691 for conjectured examples.

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) with 2 prefixed:
  2  3  7  13  19  29  37  43  53  61
  1  4  6   6  10   8   6  10   8  10
  3  2  0   4   2   2   4   2   2   2
  1  2  4   2   0   2   2   0   0   0
  1  2  2   2   2   0   2   0   0   0
  1  0  0   0   2   2   2   0   0   4
  1  0  0   2   0   0   2   0   4   2

Maple

A358692T := proc(n, k)
    option remember ;
    if n = 1 then
        if k = 1 then
            2;
        else
            ithprime(2*k-2) ;
        end if;
    else
        abs(procname(n-1, k+1)-procname(n-1, k)) ;
    end if;
end proc:
A358692 := proc(n)
    A358692T(n+1, 1) ;
end proc:
seq(A358692(n), n=1..1000) ; # R. J. Mathar, Feb 01 2023

Mathematica

z = 230; g[t_] := Abs[Differences[t]]
t = Join[{2}, Prime[2 Range[z]]]
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. A031215, A031368, A036262, A358691.

Sequence in context: A175739 A371149 A260196 * A094404 A361129 A103756

Adjacent sequences: A358689 A358690 A358691 * A358693 A358694 A358695

Keyword

nonn

Author

Clark Kimberling, Nov 27 2022

Status

approved