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Search: id:A090496
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| A090496 |
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Ratio of numerator(Bernoulli(2*n)/(2*n)) to numerator(Bernoulli(2*n)/(2*n*(2*n-1))) for n's for which they are different. |
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+0
11
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| 37, 103, 37, 59, 131, 37, 67, 37, 283, 59, 37, 101, 691, 37, 67, 37, 59, 157, 37, 617, 37, 593, 67, 59, 103, 37, 37, 37, 59, 101, 67, 157, 37, 37, 149, 233, 59, 131, 37, 37, 683, 67, 37, 271, 59, 103, 37, 37, 67, 263, 37, 59, 307, 101, 37, 37, 577, 59, 67, 37, 653, 37, 37, 59, 103, 157, 37, 67, 37, 59, 131, 101
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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A001067(n) / A046968(n) when they are different, or alternatively, gcd(A001067(n),2n-1) when that number is > 1.
These numbers always products of irregular primes (A00928).
Comment from Roland Bacher (Roland.Bacher(AT)ujf-grenoble.fr), Feb 04, 2004: All values yielding 37 are of the form 574+666*k, k=0,1,2,3,4,... and form thus an arithmetic progression with step 666=18*37=((37-1)/2)*37. All values yielding 59 are of the form 1269+1711*k, k=0,1,2,3 and 1711=28*59=((59-1)/2)*59. The two values yielding 67 are at distance 2211=((67-1)/2)*67. Conjecture: all indices yielding a given prime p form an arithmetic progression of step ((p-1)/2)*p. See A092291.
Comment from Mohammed Bouayoun, Feb 05 2004: The positions where 37 occurs appear to coincide with A026352.
Roland Bacher conjectures that values of n yielding the same quotient p form an arithmetic progression n0+d*k, where d = p(p-1)/2. Actual and conjectured values of n0 are in the sequence A092291.
Composite values do occur. An example is 2n = 272876, which yields a quotient of 37*59. This was found by tdn using the Kummer congruences and CRT: using the irregular pairs (37,32) and (59,44), we know that the following Diophantine equations must be solved for (k,l,m): 32+36*k = 44+58*l = 1+37*59*m. Some quotients are not possible, e.g. 37*67, 37*103. All quotients are the product of irregular primes A000928. Composite quotients imply there are missing terms in the arithmetic progression conjectured by Bacher. - T. D. Noe (noe(AT)sspectra.com), Feb 12 2004
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LINKS
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Eric Weisstein's World of Mathematics, Stirling's Series
Bernd Kellner, A conjecture about numerators of Bernoulli numbers
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CROSSREFS
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Cf. A090495, A001067, A046968, A092291.
Adjacent sequences: A090493 A090494 A090495 this_sequence A090497 A090498 A090499
Sequence in context: A130229 A142941 A105019 this_sequence A005107 A139934 A142051
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KEYWORD
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nonn,nice
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AUTHOR
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njas, Feb 03 2004
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EXTENSIONS
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a(1)-a(7) from Michael Somos and Edwin Clark, Feb 03, 2004.
a(8), a(9) from Robert G. Wilson v, Feb 03, 2004.
a(10)-a(12) from Eric Weisstein, Feb 03 2004
a(13)-a(39) from Cino Hilliard, Feb 03 2004
a(40)-a(44) from Eric Weisstein, Feb 04 2004
Terms from a(45) onwards from David Wasserman (wasserma(AT)spawar.navy.mil), Dec 06 2005
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