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 A001067 Numerator of Bernoulli(2*n)/(2*n). 38
 1, -1, 1, -1, 1, -691, 1, -3617, 43867, -174611, 77683, -236364091, 657931, -3392780147, 1723168255201, -7709321041217, 151628697551, -26315271553053477373, 154210205991661, -261082718496449122051, 1520097643918070802691, -2530297234481911294093 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Also numerator of "modified Bernoulli number" b(2n) = Bernoulli(2*n)/(2*n*n!). Denominators are in A057868. Ramanujan incorrectly conjectured that the sequence contains only primes (and 1) - Jud McCranie. See A112548, A119766. a(n) = A046968(n) if n < 574; a(574) = 37 * A046968(574). - Michael Somos, Feb 01 2004 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 |a(n)| is a product of powers of irregular primes (A000928), with the exception of n = 1,2,3,4,5,7. - Peter Luschny, Jul 28 2009 Conjecture: If there is a prime p such that 2*n+1 < p and p | a(n), p^2 ∤ a(n). This conjecture is true for p < 12 million. - Seiichi Manyama, Jan 21 2017 REFERENCES 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. L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979, p. 205 R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53. R. Kanigel, The Man Who Knew Infinity, pp. 91-92. J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 285. J.-P. Serre, A Course in Arithmetic, Springer-Verlag, 1973, p. 93. LINKS T. D. Noe and Seiichi Manyama, Table of n, a(n) for n = 1..314 (first 100 terms from T. D. Noe) M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 259, (6.3.18) and (6.3.19). D. Bar-Natan, T. T. Q. Le and D. P. Thurston, Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, arXiv:math/0204311 [math.QA], 2002-2003; Geometry and Topology 7-1 (2003) 1-31. G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, Integer Sequences and Periodic Points, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3. E. Z. Goren, Tables of values of Riemann zeta functions M. Janjic, Hessenberg Matrices and Integer Sequences , J. Int. Seq. 13 (2010) # 10.7.8, section 3. J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function Eric Weisstein's World of Mathematics, Eisenstein Series. Eric Weisstein's World of Mathematics, Bernoulli Number. Eric Weisstein's World of Mathematics, Modified Bernoulli Number. Wikipedia, Kummer-Vandiver conjecture FORMULA Zeta(1-2*n) = - Bernoulli(2*n)/(2*n). G.f.: numerators of coefficients of z^(2*n) in z/(exp(z)-1). - Benoit Cloitre, Jun 02 2003 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 G.f. for Bernoulli(2*n)/(2*n) = a(n)/A006953(n): (-1)^n/((2*Pi)^(2*n)*(2*n))*integral(log(1-1/t)^(2*n) dt,t=0,1). - Gerry Martens, May 18 2011 E.g.f.: a(n) = numerator((2*n+1)!*[x^(2*n+1)](1/(1-1/exp(x)))). - Peter Luschny, Jul 12 2012 |a(n)| = Numerator of Integral_{r=0..1} HurwitzZeta(1-n, r)^2. More general: |Bernoulli(2*n)| = binomial(2*n,n)*n^2*I(n) for n>=1 where I(n) denotes the integral. - Peter Luschny, May 24 2015 EXAMPLE The sequence Bernoulli(2*n)/(2*n) (n >= 1) begins 1/12, -1/120, 1/252, -1/240, 1/132, -691/32760, 1/12, -3617/8160, ... The sequence of modified Bernoulli numbers begins 1/48, -1/5760, 1/362880, -1/19353600, 1/958003200, -691/31384184832000, ... MAPLE A001067_list := proc(n) 1/(1-1/exp(z)); series(%, z, 2*n+4); seq(numer((2*i+1)!*coeff(%, z, 2*i+1)), i=0..n) end: A001067_list(21); # Peter Luschny, Jul 12 2012 MATHEMATICA Table[ Numerator[ BernoulliB[2n]/(2n)], {n, 1, 22}] (* Robert G. Wilson v, Feb 03 2004 *) PROG (PARI) {a(n) = if( n<1, 0, numerator( bernfrac(2*n) / (2*n)))}; /* Michael Somos, Feb 01 2004 */ (Sage) @CachedFunction def S(n, k) :     if k == 0 :         if n == 0 : return 1         else: return 0     return S(n, k-1) + S(n-1, n-k) def BernoulliDivN(n) :     if n == 0 : return 1     return (-1)^n*S(2*n-1, 2*n-1)/(4^n-16^n) [BernoulliDivN(n).numerator() for n in (1..22)] # Peter Luschny, Jul 08 2012 (Sage) [numerator(bernoulli(2*n)/(2*n)) for n in (1..25)] # G. C. Greubel, Sep 19 2019 (MAGMA) [Numerator(Bernoulli(2*n)/(2*n)):n in [1..40]]; // Vincenzo Librandi, Sep 17 2015 (GAP) List([1..25], n-> NumeratorRat(Bernoulli(2*n)/(2*n)));  # G. C. Greubel, Sep 19 2019 CROSSREFS Similar to but different from A046968. See A090495, A090496. Denominators given by A006953. Cf. A000367, A006863, A033563, A046968. Cf. A141590. Sequence in context: A281332 A046968 A255505 * A141590 A276594 A283301 Adjacent sequences:  A001064 A001065 A001066 * A001068 A001069 A001070 KEYWORD sign,frac,nice AUTHOR N. J. A. Sloane, Richard E. Borcherds (reb(AT)math.berkeley.edu) STATUS approved

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Last modified September 18 07:39 EDT 2020. Contains 337166 sequences. (Running on oeis4.)