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
KEYWORD
nonn
AUTHOR
T. D. Noe, Feb 26 2004
EXTENSIONS
Definition corrected by Robert Israel, Aug 20 2015
STATUS
approved