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

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

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: A333930 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