cp's OEIS Frontend

A090789 Even numbers n such that 37^2 (the square of the first irregular prime) divides the numerator of Bernoulli(n).

%I A090789 #19 Feb 16 2025 08:32:52
%S A090789 284,1184,1616,2516,2738,2948,3848,4280,5180,5476,5612,6512,6944,7844,
%T A090789 8214,8276,9176,9608,10508,10940,10952,11840,12272,13172,13604,13690,
%U A090789 14504,14936,15836,16268,16428,17168,17600,18500,18932,19166
%N A090789 Even numbers n such that 37^2 (the square of the first irregular prime) divides the numerator of Bernoulli(n).
%C A090789 Let N(n) be the numerator of the Bernoulli number B(n). This sequence is the union of three arithmetic progressions. The first, n=284+36*37*a, follows from work by Kellner on higher-order irregular pairs. In this case, the second-order pair is (37,284) because n=284 is the smallest even n such that 37^2 | N(n). The second progression, n=37(32+36*b), follows from the first-order pair (37,32). By the Kummer congruence, 37 | N(n) for n=32+36b. By a theorem of Adams, every 37th of these numbers has another factor of 37. The third progression, n=2*37^2c, yields factors of 37^2 by Adams' theorem.
%H A090789 Vincenzo Librandi, <a href="/A090789/b090789.txt">Table of n, a(n) for n = 1..1000</a>
%H A090789 Bernd Kellner, <a href="http://www.bernoulli.org/~bk/irrpairord.pdf">On irregular pairs of higher order (in German)</a>
%H A090789 S. S. Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/bernoulli/full.pdf">Prime divisors of the Bernoulli and Euler numbers</a>
%H A090789 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BernoulliNumber.html">Bernoulli Number</a>
%F A090789 These numbers are the union of three arithmetic progressions: 284 + 36*37*k, 32*37 + 36*37*k and 2*37^2*k.
%p A090789 N:= 20000: # to get all terms <= N
%p A090789 sort(convert({seq(284+36*37*k, k=0..floor((N-284)/36/37)),
%p A090789 seq(1184+36*37*k, k=0..floor((N-1184)/36/37)),
%p A090789 seq(2*37^2*k, k=1..floor(N/2/37^2))},list)); # _Robert Israel_, Aug 20 2015
%t A090789 nn=10; Union[284+36*37*Range[0, 2nn], 37(32+36*Range[0, 2nn]), 2*37^2*Range[nn]]
%Y A090789 Twice A092230.
%K A090789 nonn
%O A090789 1,1
%A A090789 _T. D. Noe_, Feb 26 2004
%E A090789 Definition corrected by _Robert Israel_, Aug 20 2015