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

1,2

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).

Links

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

Formula

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

Example

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

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 *)

Prog

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

Crossrefs

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

Cf. A326578, A326478, A326579, A326577.

Sequence in context: A194446 A251758 A250480 * A166333 A322966 A173239

Adjacent sequences: A326581 A326582 A326583 * A326585 A326586 A326587

Keyword

nonn

Author

Peter Luschny, Jul 19 2019

Status

approved