1,1

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.

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

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

nn=10; Union[284+36*37*Range[0, 2nn], 37(32+36*Range[0, 2nn]), 2*37^2*Range[nn]]

Twice A092230.

Sequence in context: A108826 A061310 A092681 this_sequence A002046 A143191 A108110

Adjacent sequences: A090786 A090787 A090788 this_sequence A090790 A090791 A090792

nonn

T. D. Noe (noe(AT)sspectra.com), Feb 26 2004