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A002445 Denominators of Bernoulli numbers B_{2n}.

%I A002445 M4189 N1746 #158 Aug 15 2025 10:28:18
%S A002445 1,6,30,42,30,66,2730,6,510,798,330,138,2730,6,870,14322,510,6,
%T A002445 1919190,6,13530,1806,690,282,46410,66,1590,798,870,354,56786730,6,
%U A002445 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 A002445 Denominators of Bernoulli numbers B_{2n}.
%C A002445 From the von Staudt-Clausen theorem, denominator(B_2n) = product of primes p such that (p-1)|2n.
%C A002445 Row products of A138239. - _Mats Granvik_, Mar 08 2008
%C A002445 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 A002445 Julius Worpitzky's 1883 algorithm for generating Bernoulli numbers is shown in A028246. - _Gary W. Adamson_, Aug 09 2008
%C A002445 There is a relation between the Euler numbers E_n and the Bernoulli numbers B_{2*n}, for n>0, namely, B_{2*n} = 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
%C A002445 a(n) is the product of all primes of the form (k + n)/(k - n). - _Thomas Ordowski_, Jul 24 2025
%D A002445 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 932.
%D A002445 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 A002445 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A002445 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002445 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A002445 See A000367 for further references and links (there are a lot).
%H A002445 T. D. Noe, <a href="/A002445/b002445.txt">Table of n, a(n) for n = 0..10000</a>
%H A002445 Amelia Bucur, José Luis López-Bonilla, and Jaime Robles-García, <a href="https://citeseerx.ist.psu.edu/document?repid=rep1&amp;type=pdf&amp;doi=e7c2ebd217850948d07d6686bf1b47b81d9047bb">A note on the Namias identity for Bernoulli numbers</a>, Journal of Scientific Research (Banaras Hindu University, Varanasi), Vol. 56 (2012), 117-120.
%H A002445 Suyuong Choi and Younghan Yoon, <a href="https://arxiv.org/abs/2508.06855">A decomposition of graph a-numbers</a>, arXiv:2508.06855 [math.CO], 2025. See p. 13.
%H A002445 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 A002445 Shizuo Kaji, Toshiaki Maeno, Koji Nuida, and Yasuhide 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 A002445 Takao Komatsu, Florian Luca, and Claudio 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 A002445 Guo-Dong Liu, H. M. Srivastava, and 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 A002445 Hong-Mei Liu, Shu-Hua Qi, and Shu-Yan 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 A002445 Romeo Meštrović, <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 A002445 Niels Nielsen, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k62119c.r=Traite+Elementaire+des+Nombres+de+Bernoulli.langFR">Traité élémentaire des nombres de Bernoulli</a>, Gauthier-Villars, 1923, pp. 398.
%H A002445 Niels Erik 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 A002445 Ronald Orozco López, <a href="https://www.researchgate.net/publication/350397609_Solution_of_the_Differential_Equation_ykeay_Special_Values_of_Bell_Polynomials_and_ka-Autonomous_Coefficients">Solution of the Differential Equation y^(k)= e^(a*y), Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients</a>, Universidad de los Andes (Colombia 2021).
%H A002445 Simon Plouffe, <a href="http://www.ibiblio.org/gutenberg/etext01/brnll10.txt">The First 498 Bernoulli numbers</a> [Project Gutenberg Etext]
%H A002445 Jan W. H. Swanepoel, <a href="https://math.colgate.edu/~integers/z50/z50.pdf">A Short Simple Probabilistic Proof of a Well Known Identity and the Derivation of Related New Identities Involving the Bernoulli Numbers and the Euler Numbers</a>, Integers (2025) Vol. 25, Art. No. A50. See p. 2.
%H A002445 <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%F A002445 E.g.f: x/(exp(x) - 1); take denominators of even powers.
%F A002445 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 A002445 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 A002445 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 A002445 a(n) = 2*denominator((2*n)!*Li_{2*n}(1)) for n > 0. - _Peter Luschny_, Jun 28 2012
%F A002445 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 A002445 a(n) = 2*A001897(n) = A027642(2*n) = 3*A277087(n) for n>0. - _Jonathan Sondow_, Dec 14 2016
%e A002445 B_{2n} = [ 1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510, ... ].
%p A002445 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 A002445 # Alternative
%p A002445 N:= 1000: # to get a(0) to a(N)
%p A002445 A:= Vector(N,2):
%p A002445 for p in select(isprime,[seq(2*i+1,i=1..N)]) do
%p A002445   r:= (p-1)/2;
%p A002445   for n from r to N by r do
%p A002445     A[n]:= A[n]*p
%p A002445   od
%p A002445 od:
%p A002445 1, seq(A[n],n=1..N); # _Robert Israel_, Nov 16 2014
%t A002445 Take[Denominator[BernoulliB[Range[0,100]]],{1,-1,2}] (* _Harvey P. Dale_, Oct 17 2011 *)
%o A002445 (PARI) a(n)=prod(p=2,2*n+1,if(isprime(p),if((2*n)%(p-1),1,p),1)) \\ _Benoit Cloitre_
%o A002445 (PARI) A002445(n,P=1)=forprime(p=2,1+n*=2,n%(p-1)||P*=p);P \\ _M. F. Hasler_, Jan 05 2016
%o A002445 (PARI) a(n) = denominator(bernfrac(2*n)); \\ _Michel Marcus_, Jul 16 2021
%o A002445 (Magma) [Denominator(Bernoulli(2*n)): n in [0..60]]; // _Vincenzo Librandi_, Nov 16 2014
%o A002445 (Sage)
%o A002445 def A002445(n):
%o A002445     if n == 0:
%o A002445         return 1
%o A002445     M = (i + 1 for i in divisors(2 * n))
%o A002445     return prod(s for s in M if is_prime(s))
%o A002445 [A002445(n) for n in (0..57)] # _Peter Luschny_, Feb 20 2016
%Y A002445 Cf. A090801 (distinct numbers appearing as denominators of Bernoulli numbers)
%Y A002445 B_n gives A027641/A027642. See A027641 for full list of references, links, formulas, etc.
%Y A002445 See A000367 for numerators. Cf. A027762, A027641, A027642, A002882, A003245, A127187, A127188, A138239, A028246, A143343, A080092, A001897, A277087.
%Y A002445 Cf. A160014 for a generalization.
%K A002445 nonn,frac,nice
%O A002445 0,2
%A A002445 _N. J. A. Sloane_