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A300711 a(n) = A000367(n)/A001067(n). 3
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..74.

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

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

CROSSREFS

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

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

Sequence in context: A052345 A197733 A241018 * A111008 A065330 A140215

Adjacent sequences:  A300708 A300709 A300710 * A300712 A300713 A300714

KEYWORD

nonn

AUTHOR

Bernd C. Kellner, Mar 11 2018

STATUS

approved

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Last modified August 18 16:13 EDT 2018. Contains 313833 sequences. (Running on oeis4.)