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%I #27 Jul 27 2019 16:32:53
%S 37,103,37,59,131,37,67,37,283,59,37,101,691,37,67,37,59,157,37,617,
%T 37,593,67,59,103,37,37,37,59,101,67,157,37,37,149,233,59,131,37,37,
%U 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
%N 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.
%C A001067(n) / A046968(n) when they are different, or alternatively, gcd(A001067(n),2n-1) when that number is > 1.
%C These numbers are always products of irregular primes (A000928).
%C 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
%C The positions where 37 occurs appear to coincide with A026352. - Mohammed Bouayoun, Feb 05 2004
%C 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.
%C 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
%H Amiram Eldar, <a href="/A090496/b090496.txt">Table of n, a(n) for n = 1..200</a> (calculated from the b-file at A090495)
%H Bernd Kellner, <a href="http://www.bernoulli.org/~bk/conjbn.pdf">A conjecture about numerators of Bernoulli numbers</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingsSeries.html">Stirling's Series</a>
%t 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 *)
%Y Cf. A090495, A001067, A046968, A092291.
%K nonn,nice
%O 1,1
%A _N. J. A. Sloane_, Feb 03 2004
%E a(1)-a(7) from _Michael Somos_ and _W. Edwin Clark_, Feb 03 2004
%E a(8), a(9) from _Robert G. Wilson v_, Feb 03 2004
%E a(10)-a(12) from _Eric W. Weisstein_, Feb 03 2004
%E a(13)-a(39) from _Cino Hilliard_, Feb 03 2004
%E a(40)-a(44) from _Eric W. Weisstein_, Feb 04 2004
%E Terms from a(45) onwards from _David Wasserman_, Dec 06 2005
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