A033563 -id:A033563 - cp's OEIS frontend

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

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

Offset: 1

N. J. A. Sloane, Richard E. Borcherds (reb(AT)math.berkeley.edu)

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

			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, ...

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.

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

Similar to but different from A046968. See A090495, A090496.
Denominators given by A006953.
Cf. A000367, A006863, A033563, A046968.
Cf. A141590, A255505.

List([1..25], n-> NumeratorRat(Bernoulli(2*n)/(2*n)));  # G. C. Greubel, Sep 19 2019

[Numerator(Bernoulli(2*n)/(2*n)):n in [1..40]]; // Vincenzo Librandi, Sep 17 2015

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

Table[ Numerator[ BernoulliB[2n]/(2n)], {n, 1, 22}] (* Robert G. Wilson v, Feb 03 2004 *)

{a(n) = if( n<1, 0, numerator( bernfrac(2*n) / (2*n)))}; /* Michael Somos, Feb 01 2004 */

@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

[numerator(bernoulli(2*n)/(2*n)) for n in (1..25)] # G. C. Greubel, Sep 19 2019

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

A112548 Numbers k such that the numerator of Bernoulli(k)/k is (apart from sign) prime.

Original entry on oeis.org

12, 16, 18, 26, 34, 36, 38, 42, 74, 114, 118, 396, 674, 1870, 4306, 22808

Offset: 1

T. D. Noe, Sep 28 2005

In 1911 Ramanujan believed that the numerator of Bernoulli(k)/k for k even was (apart from sign) always either 1 or a prime. This is false.
Equivalently, k such that the numerator of zeta(1-k) is prime. No other k < 23000. Kellner's Calcbn program was used to generate the numerators of Bernoulli(k)/k for k > 5000. Mathematica and PFGW were used to test for probable primes. David Broadhurst found n=4306, which yields a 10342-digit probable prime. For n<4306, the primes have been proved. Bouk de Water proved the prime for n=1870. All these primes are necessarily irregular.
The number generated by k=4306 was recently proved prime. See Chris Caldwell's link for more details. - T. D. Noe, Apr 06 2009
a(17) > 50000. - Robert Price, Oct 20 2013
a(17) > 74708. - Simon Plouffe, Mar 06 2022
a(17) > 270000. - Serge Batalov, Jun 26 2025

S. Ramanujan, Some properties of Bernoulli's numbers, J. Indian Math. Soc., 3 (1911), 219-234.

Bernd Kellner, Program Calcbn - A program for calculating Bernoulli numbers
Chris Caldwell, Top twenty irregular primes
K. Ono, Honoring a gift from Kumbakonam, Notices Amer. Math. Soc., 53 (2006), 640-651.
Simon Plouffe, Primes as sums of irrational numbers, Preprint, 2016.
Eric Weisstein's World of Mathematics, Irregular Prime

Cf. A001067 (numerator of Bernoulli(2n)/(2n)).
Cf. A033563 (primes in A001067).
Cf. A092132 (n such that the numerator of Bernoulli(n) is prime).
Cf. A112741 (primes p such that zeta(1-2p)/zeta(-1) is prime).
Cf. A119766.

A112548 := proc(nmax) local numr; for n from 2 to nmax by 2 do numr := abs(numer(bernoulli(n)/n)) ; if isprime(numr) then print(n) ; fi ; od ; end : A112548(3000) ; # R. J. Mathar, Jun 21 2006

Select[Range[2, 5000, 2], PrimeQ[Numerator[BernoulliB[ # ]/# ]]&]

A281331 Smallest prime factor of |A001067(n)|, or 1 if |A001067(n)| = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 691, 1, 3617, 43867, 283, 131, 103, 657931, 9349, 1721, 37, 151628697551, 26315271553053477373, 154210205991661, 137616929, 1520097643918070802691, 59, 383799511, 653, 417202699, 577, 39409, 113161, 67, 2003, 157, 1226592271, 839, 37, 688531, 3112655297839

Offset: 1

Seiichi Manyama, Jan 20 2017

nonn

			|A001067(10)| = 174611 = 283*617. So a(10) = 283.
|A001067(16)| = 7709321041217 = 37*683*305065927. So a(16) = 37.

Wikipedia, Kummer's congruences

Cf. A000928, A001067, A020639, A033563, A112548, A281332.

a[n_] := FactorInteger[Abs[Numerator[BernoulliB[2*n] / (2*n)]]][[1, 1]]; Table[a[n], {n, 1, 36}] (* Indranil Ghosh, Mar 12 2017 *)

a(n) = my(num = abs(numerator(bernfrac(2*n)/(2*n)))); if (num==1, 1, factor(num)[1,1]); \\ Michel Marcus, Jan 21 2017

a(n) = A020639(|A001067(n)|).
If n = A112548(m)/2, a(n) = |A001067(n)|.
a(18*m-2) = 37 for m > 0.

More terms from Michel Marcus, Jan 21 2017

A281332 Greatest prime factor of |A001067(n)|, or 1 if |A001067(n)| = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 691, 1, 3617, 43867, 617, 593, 2294797, 657931, 362903, 1001259881, 305065927, 151628697551, 26315271553053477373, 154210205991661, 1897170067619, 1520097643918070802691, 1798482437, 67568238839737, 153289748932447906241, 47464429777438199

Offset: 1

Seiichi Manyama, Jan 20 2017

nonn

			|A001067(10)| = 174611 = 283*617. So a(10) = 617.
|A001067(16)| = 7709321041217 = 37*683*305065927. So a(16) = 305065927.

Cf. A000928, A001067, A006530, A033563, A112548, A281331.

Table[FactorInteger[Abs@ Numerator[BernoulliB[2 n]/(2 n)]][[-1, 1]], {n, 25}] (* Michael De Vlieger, Jan 21 2017 *)

a(n) = if(abs(numerator(bernfrac(2*n) / (2*n))) == 1, 1, vecmax(factor(abs(numerator(bernfrac(2*n) / (2*n))))[,1]));
for(n=1, 25, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 11 2017

a(n) = A006530(|A001067(n)|).

If n = A112548(m)/2, a(n) = |A001067(n)|.

a(20)-a(25) from Michael De Vlieger, Jan 21 2017

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Sage

Formula

A112548 Numbers k such that the numerator of Bernoulli(k)/k is (apart from sign) prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

A281331 Smallest prime factor of |A001067(n)|, or 1 if |A001067(n)| = 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A281332 Greatest prime factor of |A001067(n)|, or 1 if |A001067(n)| = 1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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