A083687 -id:A083687 - cp's OEIS frontend

A083688 Denominator of B(2n)*H(2n)/n where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.

4, 144, 360, 33600, 15120, 34927200, 2162160, 172972800, 1543782240, 10242872640, 10346336, 2338727174784, 53542288800, 4818805992000, 3228118134040800, 1178332991611776000, 78765574305600, 12256711017694416000, 2914326249307200, 3205758874237920000, 358462128664785600

Offset: 1

Benoit Cloitre, Jun 15 2003

B(2n) is negative for even n, but this does not affect the denominator. - M. F. Hasler, Dec 24 2013

Seiichi Manyama, Table of n, a(n) for n = 1..500 (terms 1..50 from M. F. Hasler)
Ira Gessel, On Miki's identity for Bernoulli numbers J. Number Theory 110 (2005), no. 1, 75-82.

Cf. A083687.

Denominator[Table[(BernoulliB[2n]HarmonicNumber[2n])/(n (-1)^(n+1)),{n,20}]] (* Harvey P. Dale, Jun 25 2013 *)

a(n)=denominator(bernfrac(2*n)*sum(k=1,2*n,1/k)/n)

from sympy import bernoulli, harmonic
def a(n): return (bernoulli(2*n) * harmonic(2*n) / n).denominator
print([a(n) for n in range(1, 22)]) # Indranil Ghosh, Aug 04 2017

Miki's identity : B(n)*H(n)*(2/n) = sum(i=2, n-2, B(i)/i*B(n-i)/(n-i)*(1-C(n, i)))

A110841 a(n) is the number of prime factors, with multiplicity, of abs(A014509(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 2, 7, 7, 2, 2, 4, 3, 3, 7, 1, 6, 4, 5, 14, 4, 9, 5, 10, 3, 11, 2, 5, 3, 7, 11, 5, 3, 4, 15, 6, 5, 19, 10, 6, 13, 15, 5, 10, 5, 5, 6, 7, 5, 15, 7, 5, 2, 13, 4, 3, 10, 5, 9, 7, 5, 4, 9, 5, 4, 1, 7, 4, 4, 5, 3, 11, 13, 10, 5, 5, 7, 6

Offset: 0

Jonathan Vos Post, Sep 16 2005

			a(10) = 2 because A014509(10) = 529 = 23^2.
a(8) = a(19) = 1 since A014509(8) and A014509(19) are prime.

Sean A. Irvine, Table of n, a(n) for n = 0..91

Cf. A001222, A073276, A075178, A076547, A076549, A079294, A083687, A084217, A085092, A085737, A089170, A089644, A089655.

a(n) = my(b=bernfrac(2*n), c=floor(abs(b))*sign(b)); if (c==0, 0, bigomega(c)); \\ Michel Marcus, Mar 29 2020

a(n) = A001222(abs(A014509(n))).

More terms from Michel Marcus, Mar 29 2020
a(51)-a(65) from Jinyuan Wang, Apr 02 2020
More terms from Sean A. Irvine, Jul 29 2024

cp's OEIS Frontend

A083688 Denominator of B(2n)*H(2n)/n where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.

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