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 A358691 Gilbreath transform of primes p(2k-1); 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 (list; graph; refs; listen; history; text; internal format)
 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(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. Conjectured examples of Gilbreath transforms: If S = A000040 (primes), then G(S) = A000012 = (1,1,1,...) 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 (odd-indexed primes), then G(S) = A358691 = (3,3,3,3,1,1,1,...) If S = A031369, then G(S) = A358692 = (1,3,1,1,1,1,...) 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 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-1)): 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 Cf. A000040, A031368, A036262, A358692. Sequence in context: A107760 A180560 A320085 * A172368 A138070 A081334 Adjacent sequences: A358688 A358689 A358690 * A358692 A358693 A358694 KEYWORD nonn AUTHOR Clark Kimberling, Nov 27 2022 STATUS approved

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