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

Author

T. D. Noe, Feb 26 2004

Keywords

• nonn

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

Crossrefs

Twice A092230.

Programs

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]]

Formula

These numbers are the union of three arithmetic progressions: 284 + 36*37*k, 32*37 + 36*37*k and 2*37^2*k.

Extensions

Definition corrected by Robert Israel, Aug 20 2015