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A002445 Denominators of Bernoulli numbers B_2n.
(Formerly M4189 N1746)
107
1, 6, 30, 42, 30, 66, 2730, 6, 510, 798, 330, 138, 2730, 6, 870, 14322, 510, 6, 1919190, 6, 13530, 1806, 690, 282, 46410, 66, 1590, 798, 870, 354, 56786730, 6, 510, 64722, 30, 4686, 140100870, 6, 30, 3318, 230010, 498, 3404310, 6, 61410, 272118, 1410, 6, 4501770, 6, 33330, 4326, 1590, 642, 209191710, 1518, 1671270, 42 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

From the Von Staudt-Clausen theorem, denominator(B_2n) = product of primes p such that (p-1)|2n.

Row products of A138239. - Mats Granvik, Mar 08 2008

Equals row products of even rows in triangle A143343. In triangle A080092, row products = denominators of B1, B2, B4, B6, ... . - Gary W. Adamson, Aug 09 2008

Julius Worpitzky's 1883 algorithm for generating Bernoulli numbers is shown in A028246. - Gary W. Adamson, Aug 09 2008

There is a relation between the Euler numbers E_n and the Bernoulli numbers B_{2*n}, for n>0, namely, B_{2n} = A000367(n)/a(n) = ((-1)^n/(2*(1-2^{2*n})) * Sum_{k = 0..n-1} (-1)^k*2^{2*k}*C(2*n,2*k)*A000364(n-k)*A000367(k)/a(k). (See Bucur, et al.) - L. Edson Jeffery, Sep 17 2012

REFERENCES

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 136.

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

See A000367 for further references and links (there are a lot).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000

A. Bucur, J. Lopez-Bonilla, J. Robles-Garcia, A note on the Namias identity for Bernoulli numbers, Journal of Scientific Research (Banaras Hindu University, Varanasi), Vol. 56 (2012), 117-120.

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

Niels Nielsen, Traite Elementaire des Nombres de Bernoulli, Gauthier-Villars, 1923, pp. 398.

Simon Plouffe, The First 498 Bernoulli numbers [Project Gutenberg Etext]

Index entries for sequences related to Bernoulli numbers.

FORMULA

E.g.f: x/(exp(x) - 1); take denominators of even powers.

B_{2n}/(2n)! = 2*(-1)^(n-1)*(2*Pi)^(-2n) Sum_{k=1..inf} 1/k^(2n) (gives asymptotics) - Rademacher, p. 16, Eq. (9.1). In particular, B_{2*n} ~ (-1)^(n-1)*2*(2*n)!/ (2*Pi)^(2*n).

If n>=3 is prime,then a((n+1)/2)==(-1)^((n-1)/2)*12*|A000367((n+1)/2)|(mod n). - Vladimir Shevelev, Sep 04 2010

a(n)= denominator(-I*(2*n)!/(Pi*(1-2*n))*integral(log(1-1/t)^(1-2*n) dt, t=0..1)). - Gerry Martens, May 17 2011

a(n) = 2*denominator((2*n)!*Li_{2*n}(1)) for n > 0. - Peter Luschny, Jun 28 2012

EXAMPLE

B_{2n} = [ 1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510, ... ].

MAPLE

A002445 := n -> mul(i, i=select(isprime, map(i->i+1, numtheory[divisors] (2*n)))): seq(A002445(n), n=0..40); # Peter Luschny, Aug 09 2011

# Alternative

N:= 1000: # to get a(0) to a(N)

A:= Vector(N, 2):

for p in select(isprime, [seq(2*i+1, i=1..N)]) do

  r:= (p-1)/2;

  for n from r to N by r do

    A[n]:= A[n]*p

  od

od:

1, seq(A[n], n=1..N); # Robert Israel, Nov 16 2014

MATHEMATICA

Take[Denominator[BernoulliB[Range[0, 100]]], {1, -1, 2}] (* Harvey P. Dale, Oct 17 2011 *)

PROG

(PARI) a(n)=prod(p=2, 2*n+1, if(isprime(p), if((2*n)%(p-1), 1, p), 1)) \\ Benoit Cloitre

(MAGMA) [Denominator(Bernoulli(2*n)): n in [0..60]]; // Vincenzo Librandi, Nov 16 2014

CROSSREFS

Cf. A090801 (distinct numbers appearing as denominators of Bernoulli numbers)

B_n gives A027641/A027642. See A027641 for full list of references, links, formulae, etc.

See A000367 for numerators. Cf. A027762, A027641, A027642, A002882, A003245, A127187, A127188, A138239, A028246, A143343, A080092.

Cf. A160014 for a generalization.

Sequence in context: A136375 A138706 A027762 * A151711 A130512 A127662

Adjacent sequences:  A002442 A002443 A002444 * A002446 A002447 A002448

KEYWORD

nonn,frac,nice,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 24 02:09 EST 2014. Contains 249867 sequences.