A300711 - OEIS

%I #32 Jun 24 2023 06:36:55

%S 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,

%T 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,

%U 19,29,59,1,61,31,1,1,65,11,67,17,23,7,71,1,73,37

%N a(n) = A000367(n)/A001067(n).

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

%C 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.

%C 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.

%H Jeppe Stig Nielsen, <a href="/A300711/b300711.txt">Table of n, a(n) for n = 1..10000</a>

%H Bernd C. Kellner, <a href="http://dx.doi.org/10.1090/S0025-5718-06-01887-4">On irregular prime power divisors of the Bernoulli numbers</a>, Math. Comp. 76 (2007) 405-441.

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

%F a(n) * A195989(n) = n. - _Peter Luschny_, Mar 12 2018

%F From _Jianing Song_, Apr 05 2021: (Start)

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

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

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

%p A300711 := proc(n) local P, F, f, divides; divides := (a,b) -> is(irem(b,a) = 0):

%p P := 1; F := ifactors(2*n)[2]; for f in F do if not divides(f[1]-1, 2*n) then

%p P := P*f[1]^f[2] fi od; P end: seq(A300711(n), n=1..74); # _Peter Luschny_, Mar 12 2018

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

%o (Julia)

%o using Nemo

%o function A300711(n)

%o     b = bernoulli(n)

%o     div(numerator(b), numerator(b*QQ(1,n)))

%o end

%o [A300711(n) for n in 2:2:148] |> println # _Peter Luschny_, Mar 11 2018

%o (PARI) a(n) = gcd(numerator(bernfrac(2*n)), 2*n) \\ _Jianing Song_, Apr 05 2021

%o (PARI) upto(N)=bernvec(N);forstep(n=2,2*N,2,print1(gcd(numerator(bernfrac(n)), n),", ")) \\ _Jeppe Stig Nielsen_, Jun 22 2023

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

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

%K nonn

%O 1,5

%A _Bernd C. Kellner_, Mar 11 2018