A090496 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.

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

Offset: 1

Author

N. J. A. Sloane, Feb 03 2004

Keywords

• nonn
• nice

Comments

A001067(n) / A046968(n) when they are different, or alternatively, gcd(A001067(n),2n-1) 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=((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. - 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(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, Feb 12 2004

Links

• Amiram Eldar, Table of n, a(n) for n = 1..200 (calculated from the b-file at A090495)
• Bernd Kellner, A conjecture about numerators of Bernoulli numbers
• Eric Weisstein's World of Mathematics, Stirling's Series

Crossrefs

Cf. A090495, A001067, A046968, A092291.

Programs

Mathematica

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

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