A300330 -id:A300330 - cp's OEIS frontend

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

Bernd C. Kellner, Mar 11 2018

nonn

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.

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

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.

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

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

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

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

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

a(n) = gcd(numerator(bernfrac(2*n)), 2*n) \\ Jianing Song, Apr 05 2021

upto(N)=bernvec(N);forstep(n=2,2*N,2,print1(gcd(numerator(bernfrac(n)), n),", ")) \\ Jeppe Stig Nielsen, Jun 22 2023

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)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Julia

Maple

Mathematica

PARI

PARI

Formula