A300711 a(n) = A000367(n)/A001067(n).

1, 1, 1, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 7, 5, 1, 17, 1, 19, 1, 1, 11, 23, 1, 25, 13, 1, 7, 29, 1, 31, 1, 11, 17, 35, 1, 37, 19, 13, 1, 41, 1, 43, 11, 5, 23, 47, 1, 49, 1, 17, 13, 53, 1, 5, 7, 19, 29, 59, 1, 61, 31, 1, 1, 65, 11, 67, 17, 23, 7, 71, 1, 73, 37

Offset

1,5

Comments

a(n) is the trivial factor of the numerator of Bernoulli(2n) that divides 2n.

The remaining part of the (unsigned) numerator equals a product of powers of irregular primes, or 1 if and only if n = 1, 2, 3, 4, 5, 7.

Alternatively, a(n) is the product over all prime powers p^e, where p^e is the highest power of p dividing 2n and p-1 does not divide 2n.

Links

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..10000

Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007) 405-441.

Formula

a(n) = numerator(Bernoulli(2n))/numerator(Bernoulli(2n)/(2n)).

a(n) * A195989(n) = n. - Peter Luschny, Mar 12 2018

From Jianing Song, Apr 05 2021: (Start)

a(n) = gcd(numerator(Bernoulli(2n)), 2n).

a(n) = A002445(n)*(2n)/A075180(2n-1). (End)

Example

a(5) = 5, since Bernoulli(10) = 5/66 and Bernoulli(10)/10 = 1/132.

Maple

A300711 := proc(n) local P, F, f, divides; divides := (a, b) -> is(irem(b, a) = 0):
P := 1; F := ifactors(2*n)[2]; for f in F do if not divides(f[1]-1, 2*n) then
P := P*f[1]^f[2] fi od; P end: seq(A300711(n), n=1..74); # Peter Luschny, Mar 12 2018

Mathematica

Table[Numerator[BernoulliB[n]]/Numerator[BernoulliB[n]/n], {n, 2, 100, 2}]

Prog

(Julia)
using Nemo
function A300711(n)
    b = bernoulli(n)
    div(numerator(b), numerator(b*QQ(1, n)))
end
[A300711(n) for n in 2:2:148] |> println # Peter Luschny, Mar 11 2018
(PARI) a(n) = gcd(numerator(bernfrac(2*n)), 2*n) \\ Jianing Song, Apr 05 2021
(PARI) upto(N)=bernvec(N); forstep(n=2, 2*N, 2, print1(gcd(numerator(bernfrac(n)), n), ", ")) \\ Jeppe Stig Nielsen, Jun 22 2023

Crossrefs

A111008 equals the first entries and slightly differs, see a(35).

Cf. A000367, A001067, A193267, A195989, A300330, A002445, A075180.

Sequence in context: A241018 A348500 A308090 * A111008 A065330 A140215

Adjacent sequences: A300708 A300709 A300710 * A300712 A300713 A300714

Keyword

nonn

Author

Bernd C. Kellner, Mar 11 2018

Status

approved