%I A001067
%S A001067 1,1,1,1,1,691,1,3617,43867,174611,77683,236364091,657931,3392780147,
%T A001067 1723168255201,7709321041217,151628697551,26315271553053477373,
%U A001067 154210205991661,261082718496449122051,1520097643918070802691,2530297234481911294093
%V A001067 1,-1,1,-1,1,-691,1,-3617,43867,-174611,77683,-236364091,657931,-3392780147,
%W A001067 1723168255201,-7709321041217,151628697551,-26315271553053477373,
%X A001067 154210205991661,-261082718496449122051,1520097643918070802691,-2530297234481911294093
%N A001067 Numerator of Bernoulli(2n)/(2n).
%C A001067 Also numerator of "modified Bernoulli number" b(2n) = Bernoulli(2*n)/
               (2*n*n!). Denominators are in A057868.
%C A001067 Ramanujan incorrectly conjectured that the sequence contains only primes 
               (and 1) [ Jud McCranie (j.mccranie(AT)comcast.net) ]. See A112548, 
               A119766.
%C A001067 a(n)=A046968(n) if n<574; a(574)=37*A046968(574). - Michael Somos Feb 
               01 2004
%C A001067 Absolute values give denominators of constant terms of Fourier series 
               of meromorphic modular forms E_k/Delta, where E_k is the normalized 
               k th Eisenstein series [cf. Gunning or Serre references] and Delta 
               is the normalized unique weight-twelve cusp form for the full modular 
               group (the generating function of Ramanujan's tau function.) - Barry 
               Brent (barrybrent(AT)iphouse.com), Jun 01 2009
%C A001067 |a(n)| is a product of powers of irregular primes (A000928), with the 
               exeception of n = 1,2,3,4,5,7. [From Peter Luschny (peter(AT)luschny.de), 
               Jul 28 2009]
%D A001067 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, 1964 (and various 
               reprintings), p. 259, (6.3.18) and (6.3.19); also p. 810.
%D A001067 L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979, p. 205
%D A001067 R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, 
               NJ, 1962, p. 53.
%D A001067 R. Kanigel, The Man Who Knew Infinity, pp. 91-92.
%D A001067 J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, 
               p. 285.
%D A001067 J.-P. Serre, A Course in Arithmetic, Springer-Verlag, 1973, p. 93.
%H A001067 T. D. Noe, <a href="b001067.txt">Table of n, a(n) for n=1..100</a>
%H A001067 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A001067 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/
               Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</
               a>, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 
               1972, p. 259, (6.3.18) and (6.3.19).
%H A001067 D. Bar-Natan, T. T. Q. Le and D. P. Thurston, <a href="http://arXiv.org/
               abs/math.QA/0204311">Two applications of elmentary knot theory ...</
               a> Geometry and Topology 7-1 (2003) 1-31.
%H A001067 G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, <a href="http:/
               /www.cs.uwaterloo.ca/journals/JIS/index.html">Integer Sequences and 
               Periodic Points</a>, Journal of Integer Sequences, Vol. 5 (2002), 
               Article 02.2.3
%H A001067 E. Z. Goren, <a href="http://www.math.mcgill.ca/goren/ZetaValues/Riemann.html">
               Table of values of Riemann zeta function</a>
%H A001067 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RiemannZetaFunction.html">Link to a section of The World of Mathematics 
               (1).</a>
%H A001067 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               EisensteinSeries.html">Link to a section of The World of Mathematics 
               (2).</a>
%H A001067 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BernoulliNumber.html">Link to a section of The World of Mathematics 
               (3).</a>
%H A001067 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ModifiedBernoulliNumber.html">Modified Bernoulli Numbers.</a>
%H A001067 <a href="Sindx_Be.html#Bernoulli">Index entries for sequences related 
               to Bernoulli numbers.</a>
%F A001067 Zeta(1-2n) = - Bernoulli(2n)/(2n).
%F A001067 G.f.: numerators of coefficients of z^2n in z/(exp(z)-1) - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Jun 02 2003
%F A001067 For 2 <= k <= 1000 and k != 7, the 2-order of the full constant term 
               of E_k/Delta = 3 + ord_2(k - 7). - Barry Brent (barrybrent(AT)iphouse.com), 
               Jun 01 2009
%e A001067 The sequence Bernoulli(2n)/(2n) (n >= 1) begins 1/12, -1/120, 1/252, 
               -1/240, 1/132, -691/32760, 1/12, -3617/8160, ...
%e A001067 The sequence of modified Bernoulli numbers begins 1/48, -1/5760, 1/362880, 
               -1/19353600, 1/958003200, -691/31384184832000, ...
%t A001067 Table[ Numerator[ BernoulliB[2n]/(2n)], {n, 1, 22}] (from Robert G. Wilson 
               v Feb 03 2004)
%o A001067 (PARI) a(n)=if(n<1,0,numerator(bernfrac(2*n)/(2*n)))
%Y A001067 Similar to but different from A046968. See A090495, A090496.
%Y A001067 Denominators given by A006953. Cf. A000367, A033563, A006863, A046968.
%Y A001067 Cf. A141590
%Y A001067 Sequence in context: A120084 A141588 A046968 this_sequence A141590 A046988 
               A029825
%Y A001067 Adjacent sequences: A001064 A001065 A001066 this_sequence A001068 A001069 
               A001070
%K A001067 sign,frac,nice
%O A001067 1,6
%A A001067 N. J. A. Sloane (njas(AT)research.att.com), Richard E. Borcherds (reb(AT)math.berkeley.edu)