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

1,3

Comments

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

Links

Table of n, a(n) for n=1..76.

Formula

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

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 *)

Prog

(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

Crossrefs

Cf. A056612, A334958.

Sequence in context: A243146 A349440 A048671 * A205959 A318503 A328580

Adjacent sequences: A335020 A335021 A335022 * A335024 A335025 A335026

Keyword

nonn

Author

Petros Hadjicostas, May 19 2020

Status

approved