A335024 Ratios of consecutive terms of A056612.

1, 1, 2, 1, 18, 1, 4, 1, 10, 1, 12, 1, 14, 15, 8, 1, 54, 1, 100, 63, 22, 1, 8, 1, 26, 3, 28, 1, 30, 1, 16, 363, 34, 35, 36, 1, 38, 39, 40, 1, 294, 1, 4, 45, 46, 1, 48, 1, 50, 51, 52, 1, 162, 55, 56, 57, 58, 1, 60, 1, 62, 189, 32, 65, 198, 1, 68, 23, 70, 1, 24, 1, 74, 75, 76, 847, 78, 1, 80

Offset

1,3

Comments

Conjecture 1: a(n) = 1 if and only if n + 1 = p^k for some prime p and some positive integer k < p.

Conjecture 2 (due to Alois P. Heinz): a(n) = 1 <=> n+1 in A136327.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Formula

a(n) = A056612(n+1)/A056612(n).

Maple

b:= proc(n) b(n):= (1/n +`if`(n=1, 0, b(n-1))) end:
g:= proc(n) g(n):= (f-> igcd(b(n)*f, f))(n!) end:
a:= n-> g(n+1)/g(n):
seq(a(n), n=1..80);  # Alois P. Heinz, May 20 2020

Mathematica

g[n_] := GCD[n!, n! Sum[1/k, {k, 1, n}]];
a[n_] := g[n + 1]/g[n];
Array[a, 80] (* Jean-François Alcover, Dec 01 2020, after PARI *)

Prog

(PARI) g(n) = gcd(n!, n!*sum(k=1, n, 1/k)); \\ A056612
a(n) = g(n+1)/g(n); \\ Michel Marcus, May 20 2020

Crossrefs

Cf. A056612, A136327, A334958.

Sequence in context: A012895 A013077 A225768 * A270927 A089512 A300956

Adjacent sequences: A335021 A335022 A335023 * A335025 A335026 A335027

Keyword

nonn

Author

Petros Hadjicostas, May 19 2020

Status

approved