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A358692
Gilbreath transform of primes p(2k) with 2 prefixed; see Comments.
2
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.
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, Nov 27 2022
STATUS
approved