%I #34 May 14 2024 16:17:11

%S 691,12,3617,16,43867,18,283,20,617,20,131,22,593,22,103,24,2294797,

%T 24,657931,26,9349,28,362903,28,1721,30,1001259881,30,37,32,683,32,

%U 305065927,32,151628697551,34,26315271553053477373,36,154210205991661,38,137616929,40

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

%C The subsequence of irregular primes p is A046753.

%H T. D. Noe, <a href="/A189683/b189683.txt">Table of n, a(n) for n = 1..132</a>

%H Bernd C. Kellner, <a href="https://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; arXiv:<a href="https://arxiv.org/abs/math/0409223">0409223</a> [math.NT], 2004.

%H B. Mazur, <a href="https://doi.org/10.1090/S0273-0979-2011-01326-X">How can we construct abelian Galois extensions of basic number fields?</a>, Bull. Amer. Math. Soc., 48 (2011), 155-209. See footnote 64 on p. 205.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Regular_prime#Irregular_pairs">Regular prime: Irregular pairs</a>.

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

%t 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 *)

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

%K nonn

%O 1,1

%A _Jonathan Sondow_, Apr 25 2011