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 A090789 Even numbers n such that 37^2 (the square of the first irregular prime) divides the numerator of Bernoulli(n). 2
 284, 1184, 1616, 2516, 2738, 2948, 3848, 4280, 5180, 5476, 5612, 6512, 6944, 7844, 8214, 8276, 9176, 9608, 10508, 10940, 10952, 11840, 12272, 13172, 13604, 13690, 14504, 14936, 15836, 16268, 16428, 17168, 17600, 18500, 18932, 19166 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 Bernd Kellner, On irregular pairs of higher order (in German) S. S. Wagstaff, Jr., Prime divisors of the Bernoulli and Euler numbers Eric Weisstein's World of Mathematics, Bernoulli Number FORMULA These numbers are the union of three arithmetic progressions: 284 + 36*37*k, 32*37 + 36*37*k and 2*37^2*k. MAPLE N:= 20000: # to get all terms <= N sort(convert({seq(284+36*37*k, k=0..floor((N-284)/36/37)), seq(1184+36*37*k, k=0..floor((N-1184)/36/37)), seq(2*37^2*k, k=1..floor(N/2/37^2))}, list)); # Robert Israel, Aug 20 2015 MATHEMATICA nn=10; Union[284+36*37*Range[0, 2nn], 37(32+36*Range[0, 2nn]), 2*37^2*Range[nn]] CROSSREFS Twice A092230. Sequence in context: A061310 A259996 A092681 * A234970 A260087 A002046 Adjacent sequences:  A090786 A090787 A090788 * A090790 A090791 A090792 KEYWORD nonn AUTHOR T. D. Noe, Feb 26 2004 EXTENSIONS Definition corrected by Robert Israel, Aug 20 2015 STATUS approved

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Last modified November 18 02:06 EST 2019. Contains 329242 sequences. (Running on oeis4.)