

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



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



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



