A045979 Bernoulli number B_{2n} has denominator 6.

1, 7, 13, 17, 19, 31, 37, 43, 47, 49, 59, 61, 67, 71, 73, 79, 91, 97, 101, 103, 107, 109, 127, 133, 137, 139, 149, 151, 157, 163, 167, 169, 181, 193, 197, 199, 211, 217, 223, 227, 229, 241, 247, 257, 259, 263, 269, 271, 277, 283, 289

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

All primes in A053176 are in the sequence. If n is in the sequence, its factorization contains only primes in A053176. - Benoit Cloitre, Oct 19 2002

B(2n) has denominator 6 iff (n^2-1)*B(2n) is an integer. - Benoit Cloitre, Feb 15 2004

Subsequence of A156543. - Reinhard Zumkeller, Feb 10 2009

References

• B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 76.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to Bernoulli numbers.

Crossrefs

Cf. A051222, A053176, A156543.

Programs

Magma

[n: n in [0..400] | Denominator(Bernoulli(2*n)) eq 6]; // Vincenzo Librandi, Feb 06 2016

Mathematica

ok[n_] := IntegerQ[(n^2 - 1)*BernoulliB[2n]]; Select[Range[300], ok] (* Jean-François Alcover, Jun 27 2012, after Benoit Cloitre *)
result = {}; Do[count = 0;
Do[If[Not[PrimeQ[2*Divisors[n][[i]] + 1]], count++],
{i, 2, DivisorSigma[0, n]}]; If[count == DivisorSigma[0, n] - 1, AppendTo[result, n]], {n, 1, 10000}]; result  (* Richard R. Forberg, Aug 06 2016 *)
Position[BernoulliB[2 Range[300]],?(Denominator[#]==6&)]//Flatten (* _Harvey P. Dale, Jan 28 2017 *)

PARI

isok(n) = denominator(bernfrac(2*n)) == 6; \\ Michel Marcus, Feb 06 2016

Formula

a(n) seems to be asymptotic to c*n, 5 < c < 6. - Benoit Cloitre, Oct 19 2002