A189684 Number of irregular pairs (p,2n).

0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 2, 1, 2, 2, 3, 1, 1, 1, 2, 1, 4, 2, 3, 2, 4, 3, 3, 4, 3, 4, 2, 2, 4, 3, 2, 1, 2, 3, 5, 3, 5, 3, 3, 3, 3, 5, 5, 5, 5, 5, 3, 4, 3, 5, 3, 1, 4, 1, 3, 3, 7, 3, 6, 5, 3, 4, 3, 7, 6, 3, 2, 6, 6, 6, 5, 6, 5, 7, 5, 4, 7, 8, 5, 3, 2, 7

Offset: 1

Author

Jonathan Sondow, Apr 25 2011

Keywords

• nonn

Comments

a(n) is the number of primes p >= 2n+3 that divide the numerator of the Bernoulli number B_{2n}.

Links

• B. Mazur, How can we construct abelian Galois extensions of basic number fields?, Bull. Amer. Math. Soc., 48 (2011), 155-209. See footnote 64 on p. 205.
• Samual S. Wagstaff, Jr., Prime divisors of the Bernoulli and Euler numbers
• Wikipedia, Irregular pairs

Crossrefs

Cf. A000367, A000928, A035112, A046753, A189683, A189685.

Programs

Mathematica

Table[p = Select[First /@ FactorInteger[Abs[Numerator[BernoulliB[2n]]]], # >= 2*n+3 &]; Length[p], {n, 35}] (* T. D. Noe, Apr 25 2011 *)