

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
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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 higherorder irregular pairs. In this case, the secondorder 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 firstorder 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((N284)/36/37)),
seq(1184+36*37*k, k=0..floor((N1184)/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



