A358692 Gilbreath transform of primes p(2*k) 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

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. See A358691 for conjectured examples.

Examples

			Corner of successive absolute difference array (including initial row of primes p(2*k) 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
		

Links

• Index entries for sequences related to Gilbreath conjecture and transform

Crossrefs

Cf. A031215, A031368, A036262, A358691.

Programs

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}]