A001067 Numerator of Bernoulli(2*n)/(2*n).

1, -1, 1, -1, 1, -691, 1, -3617, 43867, -174611, 77683, -236364091, 657931, -3392780147, 1723168255201, -7709321041217, 151628697551, -26315271553053477373, 154210205991661, -261082718496449122051, 1520097643918070802691, -2530297234481911294093

Offset

1,6

Comments

It was incorrectly claimed that a(n) is "also numerator of "modified Bernoulli number" b(2n) = Bernoulli(2*n)/(2*n*n!)"; actually, the numerators of these fractions and the numerators of "modified Bernoulli numbers" (see A057868 for details) differ from each other and from this sequence. - Andrey Zabolotskiy, Dec 03 2022

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 divides a(n), then p^2 does not divide 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

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

Milan 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.

Wikipedia, Kummer-Vandiver conjecture

Index entries for sequences related to Bernoulli numbers

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 dr. 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, A255505.

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