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Search: id:A090997
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| A090997 |
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Numbers n such that numerator of Bernoulli number B(n) is divisible by a square. |
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6
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| 50, 98, 150, 196, 228, 242, 250, 284, 294, 338, 350, 392, 450, 484, 490, 550, 578, 650, 676, 686, 722, 726, 750, 784, 850, 914, 950, 968, 980, 1014, 1050, 1058, 1078, 1150, 1156, 1184, 1250, 1274, 1350, 1352, 1372, 1434, 1444, 1450, 1452, 1550, 1568, 1616
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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It appears that all a(n) that are divisible by p^2 and do not belong to A090943[n] are of form 2k*p^2, where p is prime and k>0 is integer. Also all numbers from A090943[n] = {228,284,914,...} are included in a(n) because they are divisible by the squares of irregular primes A094095[n] = {103,37,59,...}. Corresponding prime p such that their squares divide a(n) are listed in A090987[n] = {5,7,5,7,103,11,5,37,7,13,5,7,5,11,7,5,17,5,13,7,19,11,5,17,5,59,5,11,7,13,5,23,7,...}. - Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 19 2006
A subset of a(n) is A122270[n] = {250,686,750,1250,1372,1750,...}, numbers n such that numerator of Bernoulli number B(n) is divisible by a cube. Also a subset of a(n) is A122272[n] = {1250,3750,4802,6250,8750,9604,...}, numbers n such that numerator of Bernoulli number B(n) is divisible by p^4, where p is prime. Note that numerator of BernoulliB[6250] is divisible by 5^5. - Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 28 2006
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LINKS
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Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 28 2006, Table of n, a(n) for n = 1..152
S. S. Wagstaff, Prime factors of the absolute values of Bernoulli numerators
www.bernoulli.org, Table of factors of the numerators of Bernoulli numbers Bn in the range n=2,...,10000.
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FORMULA
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A090987[n]^2 divides numerator of BernoulliB[ a(n) ]. - Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 28 2006
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MATHEMATICA
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Mod[Numerator[BernoulliB[150]], 5^2] == 0 is True.
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CROSSREFS
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Cf. A000367, A090943. For smallest square factor see A090987.
Cf. A094095.
Cf. A122270, A122271, A122272, A122273.
Adjacent sequences: A090994 A090995 A090996 this_sequence A090998 A090999 A091000
Sequence in context: A071366 A045165 A138381 this_sequence A044139 A044520 A043220
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KEYWORD
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nonn
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AUTHOR
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Hans Havermann (pxp(AT)rogers.com), Feb 28 2004
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EXTENSIONS
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In view of the phrase "it appears", it is not clear to me that the correctness of this sequence has been rigorously established. - njas, Aug 26 2006
More terms from Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 19 2006
More terms from Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 28 2006
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