

A090496


Ratio of numerator(Bernoulli(2*n)/(2*n)) to numerator(Bernoulli(2*n)/(2*n*(2*n1))) for n's for which they are different.


11



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
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OFFSET

1,1


COMMENTS

A001067(n) / A046968(n) when they are different, or alternatively, gcd(A001067(n),2n1) when that number is > 1.
These numbers are always products of irregular primes (A000928).
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=((371)/2)*37. All values yielding 59 are of the form 1269+1711*k, k=0,1,2,3 and 1711=28*59=((591)/2)*59. The two values yielding 67 are at distance 2211=((671)/2)*67. Conjecture: all indices yielding a given prime p form an arithmetic progression of step ((p1)/2)*p. See A092291.  Roland Bacher, Feb 04 2004
The positions where 37 occurs appear to coincide with A026352.  Mohammed Bouayoun, Feb 05 2004
Roland Bacher conjectures that values of n yielding the same quotient p form an arithmetic progression n0+d*k, where d = p(p1)/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, Feb 12 2004


LINKS

Table of n, a(n) for n=1..72.
Bernd Kellner, A conjecture about numerators of Bernoulli numbers
Eric Weisstein's World of Mathematics, Stirling's Series


MATHEMATICA

A090496 = {}; Do[ r = Numerator[ b = BernoulliB[2n]/(2n) ] / Numerator[ b/(2n1) ]; If[ r > 1, Print[n, " ", r]; AppendTo[ A090496, r] ], {n, 1, 20000}]; A090496 (* JeanFrançois Alcover, Jan 24 2012 *)


CROSSREFS

Cf. A090495, A001067, A046968, A092291.
Sequence in context: A142941 A176973 A105019 * A005107 A139934 A142051
Adjacent sequences: A090493 A090494 A090495 * A090497 A090498 A090499


KEYWORD

nonn,nice


AUTHOR

N. J. A. Sloane, Feb 03 2004


EXTENSIONS

a(1)a(7) from Michael Somos and W. Edwin Clark, Feb 03 2004
a(8), a(9) from Robert G. Wilson v, Feb 03 2004
a(10)a(12) from Eric W. Weisstein, Feb 03 2004
a(13)a(39) from Cino Hilliard, Feb 03 2004
a(40)a(44) from Eric W. Weisstein, Feb 04 2004
Terms from a(45) onwards from David Wasserman, Dec 06 2005


STATUS

approved



