A393110 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.

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

Offset

2,1

Comments

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.

Links

Table of n, a(n) for n=2..15.

J. F. Colonna, La conjecture de Proth-Gilbreath, 2025.

Jean-Paul Delahaye, Nouveaux records pour la conjecture de Proth-Gilbreath, Pour la Science 580, February 2026. See table p. 72.

A. M. Odlyzko, Iterated absolute values of differences of consecutive primes, Math. Comp. 61 (1993), 373-380. See Table 2 p. 374.

Simon Plouffe, Verification of Gilbraith's conjecture up to 10^14 [sic], arXiv:2510.06688 [math.NT], 2025.

F. Proth, Sur la série des nombres premiers, Nouv. Corresp. Math., 4 (1878), 236-240.

Prog

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

Crossrefs

Cf. A000720, A036262, A006880.

Sequence in context: A233348 A061829 A063382 * A069983 A005894 A374711

Adjacent sequences: A393107 A393108 A393109 * A393111 A393112 A393113

Keyword

nonn,hard,more

Author

Michel Marcus, Feb 02 2026

Status

approved