A335023 Ratios of consecutive terms of A334958.

1, 1, 2, 1, 6, 1, 4, 3, 10, 1, 12, 1, 14, 75, 8, 1, 18, 1, 4, 21, 22, 1, 24, 5, 26, 9, 196, 1, 30, 1, 16, 33, 34, 5, 36, 1, 38, 39, 40, 1, 42, 1, 44, 45, 46, 1, 48, 7, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 60, 1, 62, 63, 32, 65, 66, 1, 68, 69, 70, 1, 72, 1, 74, 375, 76, 847

Offset: 1

Author

Petros Hadjicostas, May 19 2020

Keywords

• nonn

Comments

Conjecture: a(n) = 1 if and only if n+1 is prime.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A056612, A334958.

Programs

Maple

b:= proc(n) b(n):= (-(-1)^n/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

b[n_] := b[n] = -(-1)^n/n + If[n==1, 0, b[n-1]];
g[n_] := GCD[b[n] #, #]&[n!];
a[n_] := g[n+1]/g[n];
Array[a, 80] (* Jean-François Alcover, Nov 30 2020, after Alois P. Heinz *)

PARI

f(n) = n!*sum(k=2, n, (-1)^k/k); \\ A024168
g(n) = gcd(f(n+1), f(n)); \\ A334958
a(n) = g(n+1)/g(n); \\ Michel Marcus, May 20 2020

Formula

a(n) = A334958(n+1)/A334958(n).