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

%I A358692 #20 Mar 24 2025 04:12:28
%S A358692 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,
%T A358692 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A358692 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A358692 Gilbreath transform of primes p(2*k) with 2 prefixed; see Comments.
%C A358692 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.
%C A358692 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.
%H A358692 <a href="/index/Ge#Gilbreath">Index entries for sequences related to Gilbreath conjecture and transform</a>
%e A358692 Corner of successive absolute difference array (including initial row of primes p(2*k) with 2 prefixed):
%e A358692   2  3  7  13  19  29  37  43  53  61
%e A358692   1  4  6   6  10   8   6  10   8  10
%e A358692   3  2  0   4   2   2   4   2   2   2
%e A358692   1  2  4   2   0   2   2   0   0   0
%e A358692   1  2  2   2   2   0   2   0   0   0
%e A358692   1  0  0   0   2   2   2   0   0   4
%e A358692   1  0  0   2   0   0   2   0   4   2
%p A358692 A358692T := proc(n,k)
%p A358692     option remember ;
%p A358692     if n = 1 then
%p A358692         if k = 1 then
%p A358692             2;
%p A358692         else
%p A358692             ithprime(2*k-2) ;
%p A358692         end if;
%p A358692     else
%p A358692         abs(procname(n-1,k+1)-procname(n-1,k)) ;
%p A358692     end if;
%p A358692 end proc:
%p A358692 A358692 := proc(n)
%p A358692     A358692T(n+1,1) ;
%p A358692 end proc:
%p A358692 seq(A358692(n),n=1..1000) ; # _R. J. Mathar_, Feb 01 2023
%t A358692 z = 230; g[t_] := Abs[Differences[t]]
%t A358692 t = Join[{2}, Prime[2 Range[z]]]
%t A358692 s[1] = g[t]; s[n_] := g[s[n - 1]];
%t A358692 Table[s[n], {n, 1, z}];
%t A358692 Table[First[s[n]], {n, 1, z}]
%Y A358692 Cf. A031215, A031368, A036262, A358691.
%K A358692 nonn
%O A358692 1,2
%A A358692 _Clark Kimberling_, Nov 27 2022