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%I #21 Dec 17 2022 13:31:06
%S 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,
%T 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,
%U 55,56,19,58,1,60,1,62,21,64,13,66,1,68,23,70,1,72,1
%N a(n) = n*denominator(n*Bernoulli(n-1))/denominator(Bernoulli(n-1)).
%C 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).
%C 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.)
%H Michel Marcus, <a href="/A326478/b326478.txt">Table of n, a(n) for n = 1..10000</a>
%F a(prime(n)) = 1.
%F a(n) = n/gcd(n*N(n-1), D(n-1)), with N(k)/D(k) = B(k) the k-th Bernoulli number.
%p A326478 := n -> n*denom(n*bernoulli(n-1))/denom(bernoulli(n-1)):
%p db := n -> denom(bernoulli(n)): nb := n -> numer(bernoulli(n)):
%p a := n -> n/igcd(n*nb(n-1), db(n-1)): seq(a(n), n=1..73);
%o (PARI) a(n) = n*denominator(n*bernfrac(n-1))/denominator(bernfrac(n-1)); \\ _Michel Marcus_, Jul 17 2019
%Y Cf. A326577, A326578, A103517, A272651, A027641/A027642 (Bernoulli), A121707.
%K nonn
%O 1,4
%A _Peter Luschny_, Jul 16 2019
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