A393110 - OEIS

%I #28 Feb 05 2026 08:40:11

%S 5,15,35,65,95,135,175,248,329,417,481,635,693,800

%N a(n) = G(pi(10^n)) where pi is A000720 and G(n) is the "depth" of n in the sense of the Gilbreath-Proth conjecture.

%C Starting with the vector of the first k primes, the depth G(k) is the number of iterations required to arrive to a row whose first term is 1 and other terms are 0 and 2, when the iterations consist of taking the absolute values of the differences of the previous row.

%H J. F. Colonna, <a href="https://www.lactamme.polytechnique.fr/Mosaic/descripteurs/Proth_Gilbreath_Conjecture.01.Ang.html">La conjecture de Proth-Gilbreath</a>, 2025.

%H Jean-Paul Delahaye, <a href="https://www.pourlascience.fr/sr/logique-calcul/nombres-premiers-nouveaux-records-pour-la-conjecture-de-proth-gilbreath-28728.php">Nouveaux records pour la conjecture de Proth-Gilbreath</a>, Pour la Science 580, February 2026. See table p. 72.

%H A. M. Odlyzko, <a href="https://doi.org/10.1090/S0025-5718-1993-1182247-7">Iterated absolute values of differences of consecutive primes</a>, Math. Comp. 61 (1993), 373-380. See Table 2 p. 374.

%H Simon Plouffe, <a href="https://arxiv.org/abs/2510.06688">Verification of Gilbraith's conjecture up to 10^14 [sic]</a>, arXiv:2510.06688 [math.NT], 2025.

%H F. Proth, <a href="https://gdz.sub.uni-goettingen.de/id/PPN598948236_0004?tify=%7B%22pages%22%3A%5B271%5D%2C%22view%22%3A%22%22%7D">Sur la série des nombres premiers</a>, Nouv. Corresp. Math., 4 (1878), 236-240.

%o (PARI) G(n) = my(v=primes(n), nb=0); for (i=1, #v-1, v = vector(#v-1, k, abs(v[k+1]-v[k])); nb++; if (v[1] == 1, my(w=vector(#v-1, k, v[k+1])); if (Set(w) == Set([0,2]), return(nb)));); nb;

%o a(n) = G(primepi(10^n));

%Y Cf. A000720, A036262, A006880.

%K nonn,hard,more

%O 2,1

%A _Michel Marcus_, Feb 02 2026