A326478 - OEIS

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

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OFFSET

1,4

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

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.

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 *)

PROG