A189683 Irregular pairs (p,2k) ordered by increasing k.

691, 12, 3617, 16, 43867, 18, 283, 20, 617, 20, 131, 22, 593, 22, 103, 24, 2294797, 24, 657931, 26, 9349, 28, 362903, 28, 1721, 30, 1001259881, 30, 37, 32, 683, 32, 305065927, 32, 151628697551, 34, 26315271553053477373, 36, 154210205991661, 38, 137616929, 40

Offset: 1

Author

Jonathan Sondow, Apr 25 2011

Keywords

• nonn

Comments

The subsequence of irregular primes p is A046753.

Examples

			The first few irregular pairs are (691,12), (3617,16), (43867,18), (283,20), (617,20), (131,22), (593,22), ...
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..132
• Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007), 405-441; arXiv:0409223 [math.NT], 2004.
• 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.
• Wikipedia, Regular prime: Irregular pairs.

Crossrefs

Cf. A000367, A000928, A035112, A046753, A189684, A189685.

Programs

Mathematica

Flatten[Table[p = Select[First /@ FactorInteger[Abs[Numerator[BernoulliB[n]]]], # >= n+3 &]; Transpose[{p, Table[n, {Length[p]}]}], {n, 2, 70, 2}]] (* T. D. Noe, Apr 25 2011 *)