A111008 -id:A111008 - cp's OEIS frontend

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

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)

A164877 First bisection of A164869.

Original entry on oeis.org

0, 12, 120, 252, 240, 660, 32760, 84, 8160, 14364, 6600, 3036, 65520, 156, 24360, 429660, 16320, 204, 69090840, 228, 541200, 75852, 30360, 12972, 2227680, 3300, 82680, 43092, 48720, 20532, 3407203800, 372, 32640, 4271652, 2040, 328020, 10087262640

Offset: 0

Paul Curtz, Aug 29 2009

nonn

All entries are multiples of 12.

a(n) = A006953(n) * A111008(n), n>0.

a(n) = A164869(2n).

a(n) = A002445(n) * A005843(n).

Edited and extended by R. J. Mathar, Sep 03 2009

A228838 a(n) = n * A002445(n).

Original entry on oeis.org

0, 6, 60, 126, 120, 330, 16380, 42, 4080, 7182, 3300, 1518, 32760, 78, 12180, 214830, 8160, 102, 34545420, 114, 270600, 37926, 15180, 6486, 1113840, 1650, 41340, 21546, 24360, 10266, 1703601900, 186, 16320, 2135826, 1020, 164010, 5043631320, 222, 1140

Offset: 0

Paul Curtz, Sep 05 2013

nonn

a(n+1) is a multiple of A040031(n+1), sequence of period 2: 6, 12.
a(n) is divisible by A040879(n)=30 followed by the sequence of period 2: 6, 60. See A040214 and A165734.
Note that A164877(n) + A000367(n) = A164558(2n).

			a(0)=0*1, a(1)=1*6, a(2)=2*30=60,, a(3)=3*42=126.

Colin Barker, Table of n, a(n) for n = 0..1000

PARI
```
a(n)=n*denominator(bernfrac(2*n))
```

a(n) = A176328(2n) - A000367(n).
a(n) = A164877(n)/2.
a(n+1) = A111008(n) * A036283(n+1).
2*a(n) = A164558(2n) - A000367(n).
a(n) = A164558(2n) - A176328(2n).

Typo in data fixed by Colin Barker, Jul 03 2015

cp's OEIS Frontend

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

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A164877 First bisection of A164869.

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A228838 a(n) = n * A002445(n).

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