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

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, 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, 1, 3, 1, 5, 1, 67, 1, 3, 1, 71, 1, 73, 1, 3, 1, 1, 1, 79, 1

Offset: 1

Author

Peter Luschny, Jul 19 2019

Keywords

• nonn

Comments

Conjectures:

(1) If n > 1 then a(n) = n <=> n is prime or Carmichael (A002997).

(2) If n is odd then a(n) = 1 <=> n = 1 or is a term of A121707.

(3) The fixed points of n^2/a(n) are exactly the numbers satisfying Korselt's criterion (compare A326578 and A324050).

Examples

			a(559) =   1 and 559 is in A121707.
a(561) = 561 and 561 is Carmichael.
a(563) = 563 and 563 is prime.
		

Links

• Peter Luschny, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A002997, A121707, A027641/A027642 (Bernoulli), A324050 (Korselt).

Cf. A326578, A326478, A326579, A326577.

Programs

Maple

db := n -> denom(bernoulli(n)): nb := n -> numer(bernoulli(n)):
a := n -> igcd(n*nb(n-1), db(n-1)): seq(a(n), n=1..80);

Mathematica

a[n_] := With[{b = BernoulliB[n-1]}, GCD[n Numerator[b], Denominator[b]]];
Array[a, 80] (* Jean-François Alcover, Jul 21 2019 *)

PARI

a(n) = my(b=bernfrac(n-1)); gcd(n*numerator(b), denominator(b)); \\ Michel Marcus, Jul 19 2019

Formula

a(n) divides n, n/a(n) = A326478(n).