A326478 a(n) = n*denominator(n*Bernoulli(n-1))/denominator(Bernoulli(n-1)).

1, 1, 1, 4, 1, 6, 1, 8, 3, 10, 1, 12, 1, 14, 5, 16, 1, 18, 1, 20, 7, 22, 1, 24, 5, 26, 9, 28, 1, 30, 1, 32, 11, 34, 35, 36, 1, 38, 13, 40, 1, 42, 1, 44, 3, 46, 1, 48, 7, 50, 17, 52, 1, 54, 55, 56, 19, 58, 1, 60, 1, 62, 21, 64, 13, 66, 1, 68, 23, 70, 1, 72, 1

Offset: 1

Author

Peter Luschny, Jul 16 2019

Keywords

• nonn

Comments

Empirical: a(2*n) = [x^n] x*(2/(x - 1)^2 - 1) for n >= 1, implying the conjecture that a(2*n) = A103517(n+1) and/or A272651(n).

Conjectural, the odd fixed points > 1 of this sequence are A121707; in other words, for n > 1, denominator(n*Bernoulli(n-1)) = denominator(Bernoulli(n-1)) <=> n | Sum_{k=1..n-1} k^(n-1). (See the conjectures of Thomas Ordowski in A121707.)

Links

• Michel Marcus, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A326577, A326578, A103517, A272651, A027641/A027642 (Bernoulli), A121707.

Programs

Maple

A326478 := n -> n*denom(n*bernoulli(n-1))/denom(bernoulli(n-1)):
db := n -> denom(bernoulli(n)): nb := n -> numer(bernoulli(n)):
a := n -> n/igcd(n*nb(n-1), db(n-1)): seq(a(n), n=1..73);

Mathematica

a[n_] := Module[{b =  BernoulliB[n - 1]}, n * Denominator[n * b] / Denominator[b]]; Array[a, 100] (* Amiram Eldar, Apr 26 2024 *)

PARI

a(n) = n*denominator(n*bernfrac(n-1))/denominator(bernfrac(n-1)); \\ Michel Marcus, Jul 17 2019

Formula

a(prime(n)) = 1.

a(n) = n/gcd(n*N(n-1), D(n-1)), with N(k)/D(k) = B(k) the k-th Bernoulli number.