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%I #149 Apr 24 2024 02:46:31
%S 1,2,6,1,30,1,42,1,30,1,66,1,2730,1,6,1,510,1,798,1,330,1,138,1,2730,
%T 1,6,1,870,1,14322,1,510,1,6,1,1919190,1,6,1,13530,1,1806,1,690,1,282,
%U 1,46410,1,66,1,1590,1,798,1,870,1,354,1,56786730,1
%N Denominator of Bernoulli number B_n.
%C Row products of A138243. - _Mats Granvik_, Mar 08 2008
%C Equals row products of triangle A143343 and for a(n) > 1, row products of triangle A080092. - _Gary W. Adamson_, Aug 09 2008
%C Julius Worpitzky's 1883 algorithm for generating Bernoulli numbers is described in A028246. - _Gary W. Adamson_, Aug 09 2008
%C The sequence of denominators of B_n is defined here by convention, not by necessity. The convention amounts to mapping 0 to the rational number 0/1. It might be more appropriate to regard numerators and denominators of the Bernoulli numbers as independent sequences N_n and D_n which combine to B_n = N_n / D_n. This is suggested by the theorem of Clausen which describes the denominators as the sequence D_n = 1, 2, 6, 2, 30, 2, 42, ... which combines with N_n = 1, -1, 1, 0, -1, 0, ... to the sequence of Bernoulli numbers. (Cf. A141056 and A027760.) - _Peter Luschny_, Apr 29 2009
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 810.
%D Jacob Bernoulli, Ars Conjectandi, Basel: Thurneysen Brothers, 1713. See page 97.
%D Thomas Clausen, "Lehrsatz aus einer Abhandlung Über die Bernoullischen Zahlen", Astr. Nachr. 17 (1840), 351-352 (see P. Luschny link).
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
%D H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 230.
%D L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 137.
%D Roger Plymen, The Great Prime Number Race, AMS, 2020. See pp. 8-10.
%H T. D. Noe, <a href="/A027642/b027642.txt">Table of n, a(n) for n = 0..10000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Beáta Bényi and Péter Hajnal, <a href="https://arxiv.org/abs/1804.01868">Poly-Bernoulli Numbers and Eulerian Numbers</a>, arXiv:1804.01868 [math.CO], 2018.
%H K.-W. Chen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/CHEN/AlgBE2.html">Algorithms for Bernoulli numbers and Euler numbers</a>, J. Integer Sequences, 4 (2001), #01.1.6.
%H Ghislain R. Franssens, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Franssens/franssens13.html">On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
%H H. W. Gould and J. Quaintance, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Quaintance/quain3.html">Bernoulli Numbers and a New Binomial Transform Identity</a>, J. Int. Seq. 17 (2014) # 14.2.2
%H A. Iványi, <a href="http://www.emis.de/journals/AUSM/C5-1/math51-5.pdf">Leader election in synchronous networks</a>, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
%H M. Kaneko, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/KANEKO/AT-kaneko.html">The Akiyama-Tanigawa algorithm for Bernoulli numbers</a>, J. Integer Sequences, 3 (2000), #00.2.9.
%H 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 Peter Luschny, <a href="http://www.luschny.de/math/zeta/ClausenNumbers.htm">Generalized Clausen numbers: definition and application</a>.
%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 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha103.htm">Factorizations of many number sequences</a>
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha134.htm">Factorizations of many number sequences</a>
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha1341.htm">Factorizations of many number sequences</a>
%H A. F. Neto, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Neto/neto7.html">Carlitz's Identity for the Bernoulli Numbers and Zeon Algebra</a>, J. Int. Seq. 18 (2015) # 15.5.6.
%H Carl Pomerance and Samuel S. Wagstaff Jr, <a href="https://arxiv.org/abs/2105.13252">The denominators of the Bernoulli numbers</a>, arXiv:2105.13252 [math.NT], 2021.
%H J. Sondow and E. Tsukerman, <a href="https://arxiv.org/abs/1401.0322">The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers</a>, arXiv:1401.0322 [math.NT], 2014; see section 5.
%H Matthew Roughan, <a href="https://arxiv.org/abs/2010.09860">The Polylogarithm Function in Julia</a>, arXiv:2010.09860 [math.NA], 2020.
%H N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 5.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bernoulli_number">Bernoulli number</a>
%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F E.g.f: x/(exp(x) - 1); take denominators.
%F Let E(x) be the e.g.f., then E(x) = U(0), where U(k) = 2*k + 1 - x*(2*k+1)/(x + (2*k+2)/(1 + x/U(k+1))); (continued fraction, 3-step). - _Sergei N. Gladkovskii_, Jun 25 2012
%F E.g.f.: x/(exp(x)-1) = E(0) where E(k) = 2*k+1 - x/(2 + x/E(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 16 2013
%F E.g.f.: x/(exp(x)-1) = 2*E(0) - 2*x, where E(k)= x + (k+1)/(1 + 1/(1 - x/E(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 10 2013
%F E.g.f.: x/(exp(x)-1) = (1-x)/E(0), where E(k) = 1 - x*(k+1)/(x*(k+1) + (k+2-x)*(k+1-x)/E(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 21 2013
%F E.g.f.: conjecture: x/(exp(x)-1) = T(0)/2 - x, where T(k) = 8*k+2 + x/( 1 - x/( 8*k+6 + x/( 1 - x/T(k+1) ))); (continued fraction). - _Sergei N. Gladkovskii_, Nov 24 2013
%F a(2*n) = 2*A001897(n) = A002445(n) = 3*A277087(n) for n >= 1. _Jonathan Sondow_, Dec 14 2016
%e B_n sequence begins 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, 0, -3617/510, ...
%p (-1)^n*sum( (-1)^'m'*'m'!*stirling2(n,'m')/('m'+1),'m'=0..n);
%p A027642 := proc(n) denom(bernoulli(n)) ; end: # _Zerinvary Lajos_, Apr 08 2009
%t Table[ Denominator[ BernoulliB[n]], {n, 0, 68}] (* _Robert G. Wilson v_, Oct 11 2004 *)
%t Denominator[ Range[0, 68]! CoefficientList[ Series[x/(E^x - 1), {x, 0, 68}], x]]
%t (* Alternative code using Clausen Theorem: *)
%t A027642[k_Integer]:=If[EvenQ[k],Times@@Table[Max[1,Prime[i]*Boole[Divisible[k,Prime[i]-1]]],{i,1,PrimePi[2k]}],1+KroneckerDelta[k,1]]; (* _Enrique Pérez Herrero_, Jul 15 2010 *)
%t a[0] = 1; a[1] = 2; a[n_?OddQ] = 1; a[n_] := Times @@ Select[Divisors[n] + 1, PrimeQ]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Mar 12 2012, after Ilan Vardi, when direct computation for large n is unfeasible *)
%o (PARI) a(n)=if(n<0, 0, denominator(bernfrac(n)))
%o (PARI) a(n) = if(n == 0 || (n > 1 && n % 2), 1, vecprod(select(x -> isprime(x), apply(x -> x + 1, divisors(n))))); \\ _Amiram Eldar_, Apr 24 2024
%o (Magma) [Denominator(Bernoulli(n)): n in [0..150]]; // _Vincenzo Librandi_, Mar 29 2011
%o (Haskell)
%o a027642 n = a027642_list !! n
%o a027642_list = 1 : map (denominator . sum) (zipWith (zipWith (%))
%o (zipWith (map . (*)) (tail a000142_list) a242179_tabf) a106831_tabf)
%o -- _Reinhard Zumkeller_, Jul 04 2014
%o (Sage)
%o def A027642_list(len):
%o f, R, C = 1, [1], [1]+[0]*(len-1)
%o for n in (1..len-1):
%o f *= n
%o for k in range(n, 0, -1):
%o C[k] = C[k-1] / (k+1)
%o C[0] = -sum(C[k] for k in (1..n))
%o R.append((C[0]*f).denominator())
%o return R
%o A027642_list(62) # _Peter Luschny_, Feb 20 2016
%o (Python)
%o from sympy import bernoulli
%o [bernoulli(i).denominator() for i in range(51)] # _Indranil Ghosh_, Mar 18 2017
%Y See A027641 (numerators) for full list of references, links, formulas, etc.
%Y Cf. A002882, A003245, A127187, A127188, A138243, A028246, A143343, A080092, A141056, A027760.
%Y Cf. A242179, A106831, A000142.
%Y Cf. A001897, A002445, A277087.
%K nonn,frac,easy,core,nice,changed
%O 0,2
%A _N. J. A. Sloane_
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