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 A002445 Denominators of Bernoulli numbers B_{2n}. (Formerly M4189 N1746) 120

%I M4189 N1746

%S 1,6,30,42,30,66,2730,6,510,798,330,138,2730,6,870,14322,510,6,

%T 1919190,6,13530,1806,690,282,46410,66,1590,798,870,354,56786730,6,

%U 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

%N Denominators of Bernoulli numbers B_{2n}.

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

%C Row products of A138239. - _Mats Granvik_, Mar 08 2008

%C 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

%C Julius Worpitzky's 1883 algorithm for generating Bernoulli numbers is shown in A028246. - _Gary W. Adamson_, Aug 09 2008

%C 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

%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 932.

%D 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.

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

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

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

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

%H T. D. Noe, <a href="/A002445/b002445.txt">Table of n, a(n) for n = 0..10000</a>

%H A. Bucur, J. Lopez-Bonilla, J. Robles-Garcia, <a href="http://www.bhu.ac.in/journal/vol56-2012/BHU-11.pdf">A note on the Namias identity for Bernoulli numbers</a>, Journal of Scientific Research (Banaras Hindu University, Varanasi), Vol. 56 (2012), 117-120.

%H G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Ward/ward2.html">Integer Sequences and Periodic Points</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3

%H S. Kaji, T. Maeno, K. Nuida, Y. Numata, <a href="http://arxiv.org/abs/1506.02742">Polynomial Expressions of Carries in p-ary Arithmetics</a>, arXiv preprint arXiv:1506.02742 [math.CO], 2015.

%H T. Komatsu, F. Luca, C. de J. Pita Ruiz V., <a href="http://projecteuclid.org/euclid.pja/1398949123">A note on the denominators of Bernoulli numbers</a>, Proc. Japan Acad., 90, Ser. A (2014), p. 71-72.

%H Guo-Dong Liu, H. M. Srivastava, Hai-Quing Wang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Srivastava/sriva3.html">Some Formulas for a Family of Numbers Analogous to the Higher-Order Bernoulli Numbers</a>, J. Int. Seq. 17 (2014) # 14.4.6

%H H.-M. Liu, S-H. Qi, S.-Y. Ding, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Liu/liu4.html">Some Recurrence Relations for Cauchy Numbers of the First Kind</a>, JIS 13 (2010) # 10.3.8.

%H R. Mestrovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mestrovic/mes4.html">On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers</a>, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.

%H Niels Nielsen, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k62119c.r=Traite+Elementaire+des+Nombres+de+Bernoulli.langFR">Traite Elementaire des Nombres de Bernoulli</a>, Gauthier-Villars, 1923, pp. 398.

%H N. E. Nörlund, <a href="/A001896/a001896_1.pdf">Vorlesungen über Differenzenrechnung</a>, Springer-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463]

%H Simon Plouffe, <a href="http://www.ibiblio.org/gutenberg/etext01/brnll10.txt">The First 498 Bernoulli numbers</a> [Project Gutenberg Etext]

%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>

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

%F 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).

%F 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

%F 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

%F a(n) = 2*denominator((2*n)!*Li_{2*n}(1)) for n > 0. - _Peter Luschny_, Jun 28 2012

%F a(n) = gcd(2!S(2n+1,2),...,(2n+1)!S(2n+1,2n+1)). Here S(n,k) is the Stirling number of the second kind. See the paper of Komatsu et al. - _Istvan Mezo_, May 12 2016

%F a(n) = 2*A001897(n) = A027642(2*n) = 3*A277087(n) for n>0. - _Jonathan Sondow_, Dec 14 2016

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

%p 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

%p # Alternative

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

%p A:= Vector(N,2):

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

%p r:= (p-1)/2;

%p for n from r to N by r do

%p A[n]:= A[n]*p

%p od

%p od:

%p 1, seq(A[n],n=1..N); # _Robert Israel_, Nov 16 2014

%t Take[Denominator[BernoulliB[Range[0,100]]],{1,-1,2}] (* _Harvey P. Dale_, Oct 17 2011 *)

%o (PARI) a(n)=prod(p=2,2*n+1,if(isprime(p),if((2*n)%(p-1),1,p),1)) \\ _Benoit Cloitre_

%o (MAGMA) [Denominator(Bernoulli(2*n)): n in [0..60]]; // _Vincenzo Librandi_, Nov 16 2014

%o (PARI) A002445(n,P=1)=forprime(p=2,1+n*=2,n%(p-1)||P*=p);P \\ _M. F. Hasler_, Jan 05 2016

%o (Sage)

%o def A002445(n):

%o if n == 0: return 1

%o M = map(lambda i: i+1, divisors(2*n))

%o return mul(filter(lambda s: is_prime(s), M))

%o print [A002445(n) for n in (0..57)] # _Peter Luschny_, Feb 20 2016

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

%Y B_n gives A027641/A027642. See A027641 for full list of references, links, formulas, etc.

%Y See A000367 for numerators. Cf. A027762, A027641, A027642, A002882, A003245, A127187, A127188, A138239, A028246, A143343, A080092, A001897, A277087.

%Y Cf. A160014 for a generalization.

%K nonn,frac,nice

%O 0,2

%A _N. J. A. Sloane_

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Last modified January 16 11:32 EST 2019. Contains 319188 sequences. (Running on oeis4.)