A080209 - OEIS

%I #19 Nov 05 2025 15:22:02

%S 2,1,1,3,1,1,1,3,1,1,1,1,3,1,1,3,1,3,1,1,1,1,3,1,3,1,3,1,3,1,3,1,3,1,

%T 1,1,3,1,3,1,1,3,1,1,1,1,1,3,1,1,3,1,3,1,1,3,1,3,1,3,1,3,1,1,3,1,1,1,

%U 1,1,3,1,1,1,1,1,1,3,1,3,1,1,1,3,1,3,1,1,1,1,3,1,3,1,3,1,3,1,3,1,1,3,1,1,3

%N Gilbreath transform of the sequence of Sophie Germain primes (A005384), i.e., the diagonal of leading successive absolute differences of A005384.

%C Conjecture: The diagonal of leading successive absolute differences of the Sophie Germain primes consists, except for the initial 2, only of 1's and 3s.

%H Cristian Cobeli, Mihai Prunescu, Alexandru Zaharescu, <a href="https://arxiv.org/abs/1511.04315">A growth model based on the arithmetic Z-game</a>, arXiv:1511.04315 [math.NT], 2015.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SophieGermainPrime.html">Sophie Germain Prime.</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GilbreathsConjecture.html">Gilbreath's Conjecture</a>

%e The difference table begins:

%e    2;

%e    3,  1;

%e    5,  2,  1;

%e   11,  6,  4,  3;

%e   23, 12,  6,  2,  1;

%e   29,  6,  6,  0,  2,  1;

%t sgp[1] = Select[Prime[Range[1000]], PrimeQ[2 # + 1]&];

%t sgp[n_] := Differences[sgp[n - 1]] // Abs;

%t Table[sgp[n], {n, 1, 105}][[All, 1]] (* _Jean-François Alcover_, Feb 04 2019 *)

%Y Cf. A005384, A036262, A054977.

%K nonn

%O 1,1

%A _John W. Layman_, Mar 20 2003