|
| |
|
|
A090789
|
|
Numbers n such that 37^2 (the square of the first irregular primes) divides the numerator of Bernoulli(n).
|
|
1
| |
|
|
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
(list; graph; refs; listen; history; internal format)
|
|
|
|
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
| 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.
|
|
|
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: A108826 A061310 A092681 * A002046 A143191 A180220
Adjacent sequences: A090786 A090787 A090788 * A090790 A090791 A090792
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| T. D. Noe (noe(AT)sspectra.com), Feb 26 2004
|
| |
|
|
