|
Search: id:A002445
|
|
|
| A002445 |
|
Denominators of Bernoulli numbers B_2n.
(Formerly M4189 N1746)
|
|
+0
67
|
|
| 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
(list; graph; listen)
|
|
|
OFFSET
|
0,2
|
|
|
COMMENT
|
From the Von Staudt-Clausen theorem, denominator(B_2n) = product of primes p such that (p-1)|2n.
Row products of A138239. - Mats Granvik (mgranvik(AT)abo.fi), Mar 08 2008
Equals row products of even rows in triangle A143343. In triangle A080092, row products = denominators of B1, B2, B4, B6,... [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 09 2008]
Julius Worpitzky's 1883 algorithm for generating Bernoulli numbers is shown in A028246. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 09 2008]
|
|
REFERENCES
|
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
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.
S. Plouffe, The First 498 Bernoulli numbers [Project Gutenberg Etext]
Index entries for sequences related to Bernoulli numbers.
|
|
FORMULA
|
E.g.f: t/(e^t - 1).
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).
|
|
EXAMPLE
|
B_{2n} = [ 1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510,... ].
|
|
PROGRAM
|
(PARI) a(n)=prod(p=2, 2*n+1, if(isprime(p), if((2*n)%(p-1), 1, p), 1)) (Benoit Cloitre)
|
|
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.
Cf. also A002882, A003245, A127187, A127188.
Cf. A138239.
Cf. A028246, A143343, A080092 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 09 2008]
Sequence in context: A067879 A136375 A138706 this_sequence A027762 A151711 A130512
Adjacent sequences: A002442 A002443 A002444 this_sequence A002446 A002447 A002448
|
|
KEYWORD
|
nonn,frac,nice
|
|
AUTHOR
|
N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
Search completed in 0.003 seconds
|
