login
This site is supported by donations to The OEIS Foundation.

 

Logo

Invitation: celebrating 50 years of OEIS, 250000 sequences, and Sloane's 75th, there will be a conference at DIMACS, Rutgers, Oct 9-10 2014.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A027642 Denominator of Bernoulli number B_n. 183
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, 1, 6, 1, 870, 1, 14322, 1, 510, 1, 6, 1, 1919190, 1, 6, 1, 13530, 1, 1806, 1, 690, 1, 282, 1, 46410, 1, 66, 1, 1590, 1, 798, 1, 870, 1, 354, 1, 56786730, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Row products of A138243. - Mats Granvik, Mar 08 2008

Equals row products of triangle A143343 and for a(n)>1, row products of triangle A080092. - Gary W. Adamson, Aug 09 2008

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

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

REFERENCES

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.

Clausen, Thomas, "Lehrsatz aus einer Abhandlung Über die Bernoullischen Zahlen", Astr. Nachr. 17 (1840), 351-352 (see P. Luschny link).

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.

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.

A. Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.

L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 137.

LINKS

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

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

K.-W. Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.

Hisanori Mishima, Factorizations of many number sequences

Hisanori Mishima, Factorizations of many number sequences

Hisanori Mishima, Factorizations of many number sequences

P. Luschny, Generalized Clausen numbers: definition and application.

J. Sondow and E. Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 5.

Wikipedia, Bernoulli number

Index entries for sequences related to Bernoulli numbers.

Index entries for "core" sequences

FORMULA

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

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

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

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

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

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

EXAMPLE

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, ...

MAPLE

(-1)^n*sum( (-1)^'m'*'m'!*stirling2(n, 'm')/('m'+1), 'm'=0..n);

A027642 := proc(n) denom(bernoulli(n)) ; end: # Zerinvary Lajos, Apr 08 2009

MATHEMATICA

Table[ Denominator[ BernoulliB[n]], {n, 0, 68}] (* Robert G. Wilson v, Oct 11 2004 *)

Denominator[ Range[0, 68]! CoefficientList[ Series[x/(E^x - 1), {x, 0, 68}], x]]

(* Alternative code using Clausen Theorem: *)

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 *)

a[0] = 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 *)

PROG

(PARI) a(n)=if(n<0, 0, denominator(bernfrac(n)))

(MAGMA) [Denominator(Bernoulli(n)): n in [0..150]]; // Vincenzo Librandi, Mar 29 2011

(Haskell)

a027642 n = a027642_list !! n

a027642_list = 1 : map (denominator . sum) (zipWith (zipWith (%))

   (zipWith (map . (*)) (tail a000142_list) a242179_tabf) a106831_tabf)

-- Reinhard Zumkeller, Jul 04 2014

CROSSREFS

See A027641 (numerators) for full list of references, links, formulae, etc.

Cf. A002882, A003245, A127187, A127188, A138243, A028246, A143343, A080092, A141056, A027760.

Cf. A242179, A106831, A000142.

Sequence in context: A221913 A180512 A132181 * A117214 A185972 A182918

Adjacent sequences:  A027639 A027640 A027641 * A027643 A027644 A027645

KEYWORD

nonn,frac,easy,core,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified September 2 12:55 EDT 2014. Contains 246357 sequences.