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A326584 a(n) = gcd(n*N(n-1), D(n-1)), with N(n)/D(n) = B(n) the n-th Bernoulli number.

%I A326584 #29 Oct 06 2021 03:51:57
%S A326584 1,2,3,1,5,1,7,1,3,1,11,1,13,1,3,1,17,1,19,1,3,1,23,1,5,1,3,1,29,1,31,
%T A326584 1,3,1,1,1,37,1,3,1,41,1,43,1,15,1,47,1,7,1,3,1,53,1,1,1,3,1,59,1,61,
%U A326584 1,3,1,5,1,67,1,3,1,71,1,73,1,3,1,1,1,79,1
%N A326584 a(n) = gcd(n*N(n-1), D(n-1)), with N(n)/D(n) = B(n) the n-th Bernoulli number.
%C A326584 Conjectures:
%C A326584 (1) If n > 1 then a(n) = n <=> n is prime or Carmichael (A002997).
%C A326584 (2) If n is odd then a(n) = 1 <=> n = 1 or is a term of A121707.
%C A326584 (3) The fixed points of n^2/a(n) are exactly the numbers satisfying Korselt's criterion (compare A326578 and A324050).
%H A326584 Peter Luschny, <a href="/A326584/b326584.txt">Table of n, a(n) for n = 1..10000</a>
%F A326584 a(n) divides n, n/a(n) = A326478(n).
%e A326584 a(559) =   1 and 559 is in A121707.
%e A326584 a(561) = 561 and 561 is Carmichael.
%e A326584 a(563) = 563 and 563 is prime.
%p A326584 db := n -> denom(bernoulli(n)): nb := n -> numer(bernoulli(n)):
%p A326584 a := n -> igcd(n*nb(n-1), db(n-1)): seq(a(n), n=1..80);
%t A326584 a[n_] := With[{b = BernoulliB[n-1]}, GCD[n Numerator[b], Denominator[b]]];
%t A326584 Array[a, 80] (* _Jean-François Alcover_, Jul 21 2019 *)
%o A326584 (PARI) a(n) = my(b=bernfrac(n-1)); gcd(n*numerator(b), denominator(b)); \\ _Michel Marcus_, Jul 19 2019
%Y A326584 Cf. A000040, A002997, A121707, A027641/A027642 (Bernoulli), A324050 (Korselt).
%Y A326584 Cf. A326578, A326478, A326579, A326577.
%K A326584 nonn
%O A326584 1,2
%A A326584 _Peter Luschny_, Jul 19 2019