A141056 -id:A141056 - cp's OEIS frontend

A176591 Bernoulli denominators A141056(n), with the exception a(1)=1.

1, 1, 6, 2, 30, 2, 42, 2, 30, 2, 66, 2, 2730, 2, 6, 2, 510, 2, 798, 2, 330, 2, 138, 2, 2730, 2, 6, 2, 870, 2, 14322, 2, 510, 2, 6, 2, 1919190, 2, 6, 2, 13530, 2, 1806, 2, 690, 2, 282, 2, 46410, 2, 66, 2, 1590, 2, 798, 2, 870, 2, 354, 2, 56786730, 2, 6, 2, 510, 2, 64722, 2, 30, 2, 4686, 2, 140100870, 2, 6, 2, 30, 2

Offset: 0

Paul Curtz, Apr 21 2010

These are also the denominators of a sequence generated by inverse binomial transform of a modified Bernoulli sequence described in (with numerators in) A176328.

Antti Karttunen, Table of n, a(n) for n = 0..4096

Cf. A141056, A160014.

read("transforms") ; evb := [1, 0, seq(bernoulli(n), n=2..50)] ; BINOMIALi(evb) ; apply(denom, %) ; # R. J. Mathar, Dec 01 2010
seq(denom((bernoulli(i,1)+bernoulli(i,2))/2),i=0..50); # Peter Luschny, Jun 17 2012

a[n_] := If[OddQ[n], 2, BernoulliB[n] // Denominator]; a[1] = 1; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 29 2012 *)
Join[{1,1},BernoulliB[Range[2,80]]/.(0->1/2)//Denominator] (* Harvey P. Dale, Dec 31 2018 *)

A176591(n) = { my(p=1); if(n>1, fordiv(n, d, my(r=d+1); if(isprime(r), p = p*r))); return(p); }; \\ Antti Karttunen, Dec 20 2018, after code in A141056

a(n) = A141056(n), n <> 1.
a(n) = A027760(n), n>1.
a(2n) = A002445(n), a(2n+1)= A040000(n).

More terms from Antti Karttunen, Dec 20 2018

A213621 The denominator of the Bernoulli polynomial B(n,x) divided by the Clausen number C(n), A144845(n)/A141056(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 5, 1, 3, 1, 105, 5, 3, 1, 15, 5, 105, 7, 165, 5, 15, 1, 273, 7, 7, 1, 15, 1, 231, 77, 1785, 35, 3, 1, 25935, 455, 105, 7, 1155, 55, 1155, 7, 2415, 35, 105, 1, 3315, 221, 429, 11, 165, 55, 399, 19, 435, 5, 15, 1, 465465, 5005, 2145

Offset: 0

Peter Luschny, Jun 16 2012

nonn

Cf. A213623.

# Clausen(n,k) defined in A160014.
seq(denom(bernoulli(i,x))/Clausen(i,1), i=0..63);

c[0, ] = 1; c[n, k_] := Times @@ (Select[Divisors[n], PrimeQ[#+k]&] + k);
Table[Denominator[BernoulliB[i, x] // Together]/c[i, 1], {i, 0, 63}] (* Jean-François Alcover, Aug 02 2019 *)

A213623 Numbers n such that the denominator of the Bernoulli polynomial B(n,x) equals the Clausen number C(n), {n | A144845(n) = A141056(n)}.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 10, 12, 16, 24, 28, 30, 36, 48, 60, 120

Offset: 0

Peter Luschny, Jun 16 2012

Is this a finite sequence?

Cf. A141056, A144845, A213621.

# Clausen(n, k) defined in A160014.
seq(`if`(denom(bernoulli(i,x))=Clausen(i,1),i,NULL), i=0..120);

Clausen[n_, k_] := If[n == 0, 1, Times @@ (Select[Divisors[n], PrimeQ[# + k]&] + k)];
Select[Range[0, 120], Denominator[BernoulliB[#, x] // Together] == Clausen[#, 1]&] (* Jean-François Alcover, Aug 13 2019 *)

A027642 Denominator of Bernoulli number B_n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Row products of A138243. - Mats Granvik, Mar 08 2008
From Gary W. Adamson, Aug 09 2008: (Start)
Equals row products of triangle A143343 and for a(n) > 1, row products of triangle A080092.
Julius Worpitzky's 1883 algorithm for generating Bernoulli numbers is described in A028246. (End)
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

			The sequence of Bernoulli numbers B_n (n = 0, 1, 2, ...) 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, ... [Clarified by _N. J. A. Sloane_, Jun 02 2025]

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.
Jacob Bernoulli, Ars Conjectandi, Basel: Thurneysen Brothers, 1713. See page 97.
Thomas Clausen, "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.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 106-108.
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.
L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 137.
Roger Plymen, The Great Prime Number Race, AMS, 2020. See pp. 8-10.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 161.

T. D. Noe, Table of n, a(n) for n = 0..10000
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Beáta Bényi and Péter Hajnal, Poly-Bernoulli Numbers and Eulerian Numbers, arXiv:1804.01868 [math.CO], 2018.
Kwang-Wu Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.
Bakir Farhi, Formulas Involving Bernoulli and Stirling Numbers of Both Kinds, Journal of Integer Sequences, Vol. 28 (2025), Article 25.2.6. See p. 16.
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.
H. W. Gould and Jocelyn Quaintance, Bernoulli Numbers and a New Binomial Transform Identity, J. Int. Seq. 17 (2014) # 14.2.2
Antal Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
Guo-Dong Liu, H. M. Srivastava, and Hai-Quing Wang, Some Formulas for a Family of Numbers Analogous to the Higher-Order Bernoulli Numbers, J. Int. Seq. 17 (2014) # 14.4.6
Peter Luschny, Generalized Clausen numbers: definition and application.
Romeo Meštrović, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Antônio Francisco Neto, Carlitz's Identity for the Bernoulli Numbers and Zeon Algebra, J. Int. Seq. 18 (2015) # 15.5.6.
Ezgi Polat and Yilmaz Simsek, New formulas for Bernoulli polynomials with applications of matrix equations and Laplace transform, Pub. de l'Inst. Math. (2024) Vol. 116, Issue 130, 59-74. See p. 69.
Carl Pomerance and Samuel S. Wagstaff Jr, The denominators of the Bernoulli numbers, arXiv:2105.13252 [math.NT], 2021.
Jonathan Sondow and Emmanuel Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 5.
Matthew Roughan, The Polylogarithm Function in Julia, arXiv:2010.09860 [math.NA], 2020.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
Wikipedia, Bernoulli number
Index entries for sequences related to Bernoulli numbers.
Index entries for "core" sequences

See A027641 (numerators) for full list of references, links, formulas, etc.
Cf. A002882, A003245, A127187, A127188, A138243, A028246, A143343, A080092, A141056, A027760.
Cf. A242179, A106831, A000142.
Cf. A001897, A002445, A277087.

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

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

(-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

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[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 *)

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

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

from sympy import bernoulli
[bernoulli(i).denominator for i in range(51)] # Indranil Ghosh, Mar 18 2017

def A027642_list(len):
    f, R, C = 1, [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        f *= n
        for k in range(n, 0, -1):
            C[k] = C[k-1] / (k+1)
        C[0] = -sum(C[k] for k in (1..n))
        R.append((C[0]*f).denominator())
    return R
A027642_list(62) # Peter Luschny, Feb 20 2016

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
a(2*n) = 2*A001897(n) = A002445(n) = 3*A277087(n) for n >= 1. Jonathan Sondow, Dec 14 2016

A160014 Generalized Clausen numbers (table read by antidiagonals).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 2, 2, 3, 1, 1, 5, 30, 15, 5, 5, 1, 6, 2, 3, 1, 5, 1, 1, 7, 42, 21, 35, 35, 7, 7, 1, 2, 2, 15, 1, 5, 1, 7, 1, 1, 3, 30, 3, 5, 5, 7, 7, 1, 1, 1, 10, 2, 3, 1, 35, 1, 7, 1, 1, 1, 1, 11, 66, 165, 385, 55, 77, 77, 11, 11, 11, 11, 1

Offset: 0

Peter Luschny, Apr 29 2009

T(n,k) = Product_{ p - k | n} p, where p is prime.
T(n,0) is the squarefree kernel of n (A007947).
T(n,1) are the classical Clausen numbers (A141056). The classical Clausen numbers are by the von Staudt-Clausen theorem the denominators of the Bernoulli numbers.

			[k\n][0--1--2---3---4---5---6---7----8----9---10---11----12---13---14----15]
[0]...1..1..2...3...2...5...6...7....2....3...10...11.....6...13...14....15
[1]...1..2..6...2..30...2..42...2...30....2...66....2..2730....2....6.....2
[2]...1..3..3..15...3..21..15...3....3..165...21...39....15....3....3..1785
[3]...1..1..5...1..35...1...5...1..385....1...65....1....35....1...85.....1
[4]...1..5..5..35...5...5..35..55....5..455....5....5....35...85...55...665
[5]...1..1..7...1...7...1..77...1...91....1....7....1..1309....1..133.....1
T(3,4) = 35 = 5*7 because 5 and 7 are the only prime numbers p such that
(p - 4) divides 3.

Clausen, Thomas, "Lehrsatz aus einer Abhandlung ueber die Bernoullischen Zahlen", Astr. Nachr. 17 (1840), 351-352.

Charles R Greathouse IV, Rows n = 0..100, flattened
A. Hurwitz, Über die Entwicklungskoeffizienten der lemniskatischen Funktionen, Math. Ann., 51 (1899), 196-226; Mathematische Werke. Vols. 1 and 2, Birkhäuser, Basel, 1962-1963, see Vol. 2, No. LXVII.
Peter Luschny, Generalized Bernoulli numbers.

Cf. A007947, A141056, A027760, A027642.

Clausen := proc(n,k) local S,i;
S := numtheory[divisors](n);
S := map(i->i+k,S);
S := select(isprime,S);
mul(i,i=S) end:

t[0, ] = 1; t[n, k_] := Times @@ (Select[Divisors[n], PrimeQ[# + k] &] + k); Table[t[n-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 26 2013 *)

T(n,k)=if(n,my(s=1);fordiv(n,d,if(isprime(d+k),s*=d+k)); s, 1)
for(s=0,9,for(k=0,s,print1(T(s-k,k)", "))) \\ Charles R Greathouse IV, Jun 26 2013

def Clausen(n, k):
    if k == 0: return 1
    return mul(filter(lambda s: is_prime(s), map(lambda i: i+n, divisors(k))))
for n in (0..5): [Clausen(n, k) for k in (0..15)]   # Peter Luschny, Jun 05 2013

Swapped n<>k fixed by Peter Luschny, May 04 2009

A027760 Denominator of Sum_{p prime, p-1 divides n} 1/p.

Original entry on oeis.org

2, 6, 2, 30, 2, 42, 2, 30, 2, 66, 2, 2730, 2, 6, 2, 510, 2, 798, 2, 330, 2, 138, 2, 2730, 2, 6, 2, 870, 2, 14322, 2, 510, 2, 6, 2, 1919190, 2, 6, 2, 13530, 2, 1806, 2, 690, 2, 282, 2, 46410, 2, 66, 2, 1590, 2, 798, 2, 870, 2, 354, 2, 56786730, 2, 6, 2, 510, 2

Offset: 1

N. J. A. Sloane

The GCD of integers x^(n+1)-x, for all integers x. - Roger Cuculiere (cuculier(AT)imaginet.fr), Jan 19 2002
If each x in a ring satisfies x^(n+1)=x, the characteristic of the ring is a divisor of a(n) (Rosenblum 1977). - Daniel M. Rosenblum (DMRosenblum(AT)world.oberlin.edu), Sep 24 2008
The denominators of the Bernoulli numbers for n>0. B_n sequence begins 1, -1/2, 1/6, 0/2, -1/30, 0/2, 1/42, 0/2, ... This is an alternative version of A027642 suggested by the theorem of Clausen. To add a(0) = 1 has been proposed in A141056. - Peter Luschny, Apr 29 2009
For N > 1, a(n) is the greatest number k such that x*y^n ==y*x^n (mod k) for any integers x and y.  Example: a(19) = 798 because x*y^19 ==y*x^19 (mod 798). - Michel Lagneau, Apr 21 2012
a(n) is the largest k such that b^(n+1) == b (mod k) for every integer b. - Mateusz Szymański, Feb 18 2016, corrected by Thomas Ordowski, Jul 01 2018
When n is even, a(n) is the product of the distinct primes dividing the denominator of zeta(1-n), where zeta(s) is the Riemann zeta function. - Griffin N. Macris, Jun 13 2016
If n+1 is prime, then A002322(a(n)) = n. Composite numbers n+1 such that A002322(a(n)) = n are in A317210. - Max Alekseyev and Thomas Ordowski, Jul 09 2018

			1/2, 5/6, 1/2, 31/30, 1/2, 41/42, 1/2, 31/30, 1/2, 61/66, 1/2, 3421/2730, 1/2, 5/6, 1/2, 557/510, ...

T. D. Noe, Table of n, a(n) for n = 1..10000
Thomas Clausen, Lehrsatz aus einer Abhandlung Über die Bernoullischen Zahlen, Astr. Nachr. 17 (1840), 351-352. [_Peter Luschny_, Apr 29 2009]
S. C. Locke and A. Mandel, Problem E 2901, American Mathematical Monthly 88 (1981), p. 538. Solution in Vol. 90 (1983), pp. 212-213. [Daniel M. Rosenblum (DMRosenblum(AT)world.oberlin.edu), Jul 31 2008]
D. M. Rosenblum, Problem 1019, Mathematics Magazine 50 (1977), p. 164. Solution by T. Orloff in Vol. 52 (1979), p. 50.
Wikipedia, Bernoulli number

Cf. A002997, A027759, A027642, A141056, A317210.

A027760 := proc(n) local s,p; s := 0 ; p := 2; while p <= n+1 do if n mod (p-1) = 0 then s := s+1/p; fi; p := nextprime(p) ; od: denom(s) ; end: # R. J. Mathar, Aug 12 2008

clausen[n_] := Product[i, {i, Select[ Map[ # + 1 &, Divisors[n]], PrimeQ]}]
Table[clausen[i], {i, 1, 20}] (* Peter Luschny, Apr 29 2009 *)
f[n_] := Times @@ Select[Divisors@n + 1, PrimeQ]; Array[f, 56] (* Robert G. Wilson v, Apr 25 2012 *)

a(n)=denominator(sumdiv(n,d,if(isprime(d+1),1/(d+1)))) \\ Charles R Greathouse IV, Jul 08 2011

a(n)=my(pr=1);fordiv(n,d,if(isprime(d+1),pr*=d+1));pr \\ Charles R Greathouse IV, Jul 08 2011

def A027760(n):
    return mul(filter(lambda s: is_prime(s), map(lambda i: i+1, divisors(n))))
[A027760(n) for n in (1..56)]  # Peter Luschny, May 23 2013

a(2*k) = A091137(2*k)/A091137(2*k-1). - Paul Curtz, Aug 05 2008
a(n) = product_{p prime, p-1 divides n}. - Eric M. Schmidt, Aug 01 2013
a(2n-1) = 2. - Robert G. Wilson v, Jul 23 2018

Formula submitted with A141417 added by R. J. Mathar, Nov 17 2010

A176327 Numerators of the rational sequence with e.g.f. (x/2)*(1+exp(-x))/(1-exp(-x)).

Original entry on oeis.org

1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, -174611, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -7709321041217, 0, 2577687858367, 0, -26315271553053477373, 0, 2929993913841559, 0, -261082718496449122051

Offset: 0

Paul Curtz, Apr 15 2010

sign

Numerator of the Bernoulli number B_n, except B(1)=0.
A027641 is the main entry for this sequence, which is only a minor variation. - N. J. A. Sloane, Nov 29 2010.
This could formally be defined by building the arithmetic mean of the numerators in A164555(n) and A027641(n).

Antti Karttunen, Table of n, a(n) for n = 0..200

Cf. A176289 (denominators), A027642, A141056, A164020, A165823

seq(numer((bernoulli(i,0)+bernoulli(i,1))/2),i=0..40); # Peter Luschny, Jun 17 2012

terms = 41; egf = (x/2)*((1 + Exp[-x])/(1 - Exp[-x])) + O[x]^(terms+1);
CoefficientList[egf, x]*Range[0, terms-1]! // Numerator (* Jean-François Alcover, Jun 13 2017 *)

apply(numerator, Vec(serlaplace((x/2)*(1+exp(-x))/(1-exp(-x))))) \\ Charles R Greathouse IV, Sep 26 2017

a(2n+1) = 0. a(2n ) = A000367(n).

a(n) = A164555(n) = A027641(n) if n <>1.

New name from Peter Luschny, Jun 18 2012

A141459 a(n) = Product_{p-1 divides n} p, where p is an odd prime.

Original entry on oeis.org

1, 1, 3, 1, 15, 1, 21, 1, 15, 1, 33, 1, 1365, 1, 3, 1, 255, 1, 399, 1, 165, 1, 69, 1, 1365, 1, 3, 1, 435, 1, 7161, 1, 255, 1, 3, 1, 959595, 1, 3, 1, 6765, 1, 903, 1, 345, 1, 141, 1, 23205, 1, 33, 1, 795, 1, 399, 1, 435, 1, 177, 1, 28393365, 1, 3, 1, 255, 1, 32361, 1, 15, 1, 2343, 1, 70050435

Offset: 0

Paul Curtz, Aug 08 2008

nonn

Previous name was: A027760(n)/2 for n>=1, a(0) = 1.
Conjecture: a(n) = denominator of integral_{0..1}(log(1-1/x)^n) dx. - Jean-François Alcover, Feb 01 2013
Define the generalized Bernoulli function as B(s,z) = -s*z^s*HurwitzZeta(1-s,1/z) for Re(1/z) > 0 and B(0,z) = 1 for all z; further the generalized Bernoulli polynomials as Bp(m,n,z) = Sum_{j=0..n} B(j,m)*C(n,j)*(z-1)^(n-j) then the a(n) are denominators of Bp(2,n,1), i. e. of the generalized Bernoulli numbers in the case m=2. The numerators of these numbers are A157779(n). - Peter Luschny, May 17 2015
From Peter Luschny, Nov 22 2015: (Start)
a(n) are the denominators of the centralized Bernoulli polynomials 2^n*Bernoulli(n, x/2+1/2) evaluated at x=1. The numerators are A239275(n).
a(n) is the odd part of A141056(n).
a(n) is squarefree, by the von Staudt-Clausen theorem. (End)
Apparently a(n) = denominator(Sum_{k=0..n-1}(-1)^k*E2(n-1, k+1)/binomial(2*n-1, k+1)) where E2(n, k) denotes the second-order Eulerian numbers A340556. - Peter Luschny, Feb 17 2021

			The denominators of 1, 0, -1/3, 0, 7/15, 0, -31/21, 0, 127/15, 0, -2555/33, 0, 1414477/1365, ...

Robert Israel, Table of n, a(n) for n = 0..10000
Peter Luschny, Generalized Bernoulli Numbers and Polynomials

Cf. A027760, A141056, A141459, A157779, A157818, A160014, A226157, A239275, A340556.

Bfun := (s,z) -> `if`(s=0,1,-s*z^s*Zeta(0,1-s,1/z): # generalized Bernoulli function
Bpoly := (m,n,z) -> add(Bfun(j,m)*binomial(n,j)*(z-1)^(n-j),j=0..n): # generalized Bernoulli polynomials
seq(Bpoly(2,n,1),n=0..50): denom([%]);
# which simplifies to:
a := n -> 0^n+add(Zeta(1-j)*(2^j-2)*j*binomial(n,j), j=1..n):
seq(denom(a(n)), n=0..50); # Peter Luschny, May 17 2015
# Alternatively:
with(numtheory):
ClausenOdd := proc(n) local S, m;
S := map(i -> i + 1, divisors(n));
S := select(isprime, S) minus {2};
mul(m, m = S) end: seq(ClausenOdd(n), n=0..72); # Peter Luschny, Nov 22 2015
# Alternatively:
N:= 1000: # to get a(0) to a(N)
V:= Array(0..N, 1):
for p in select(isprime, [seq(i,i=3..N+1,2)]) do
  R:=[seq(j,j=p-1..N, p-1)]:
  V[R]:= V[R] * p;
od:
convert(V,list); # Robert Israel, Nov 22 2015

a[n_] := If[OddQ[n], 1, Denominator[-2*(2^(n - 1) - 1)*BernoulliB[n]]]; Table[a[n], {n, 0, 72}] (* Jean-François Alcover, Jan 30 2013 *)
Table[Times @@ Select[Divisors@ n + 1, PrimeQ@ # && OddQ@ # &] + Boole[n == 0], {n, 0, 72}] (* Michael De Vlieger, Apr 30 2017 *)

A141056(n) =
{
    p = 1;
    if (n > 0,
        fordiv(n, d,
            r = d + 1;
            if (isprime(r) & r>2, p = p*r)
        )
    );
    return(p)
}
for(n=0, 72, print1(A141056(n), ", ")); \\ Peter Luschny, Nov 22 2015

def A141459_list(size):
    f = x / sum(x^(n*2+1)/factorial(n*2+1) for n in (0..2*size))
    t = taylor(f, x, 0, size)
    return [(factorial(n)*s).denominator() for n,s in enumerate (t.list())]
print(A141459_list(72)) # Peter Luschny, Jul 05 2016

a(2*n+1) = 1. a(2*n)= A001897(n).
a(n) = denominator(0^n + Sum_{j=1..n} zeta(1-j)*(2^j-2)*j*C(n,j)). - Peter Luschny, May 17 2015
Let P(x)= Sum_{n>=0} x^(2*n+1)/(2*n+1)! then a(n) = denominator( n! [x^n] x/P(x) ). - Peter Luschny, Jul 05 2016
a(n) = A157818(n)/4^n. See a comment under A157817, also for other Bernoulli numbers B[4,1] and B[4,3] with this denominator. - Wolfdieter Lang, Apr 28 2017

1 prepended and offset set to 0 by Peter Luschny, May 17 2015

New name from Peter Luschny, Nov 22 2015

A176328 Numerators of the rational sequence with e.g.f. (x/2)*(exp(-x) + 1)/(exp(x) - 1).

Original entry on oeis.org

1, -1, 7, -3, 59, -5, 127, -7, 119, -9, 335, -11, 15689, -13, 49, -15, 463, -17, 51049, -19, -171311, -21, 856031, -23, -236331331, -25, 8553181, -27, -23749448849, -29, 8615841490835, -31, -7709321033057, -33, 2577687858469

Offset: 0

Paul Curtz, Apr 15 2010

Numerator of the n-term of the inverse binomial transform of the modified Bernoulli sequence A176327(k)/A027642(k).
The sequence of modified Bernoulli numbers A176327(k)/A027642(k) is defined to be the same as the Bernoulli sequence, except the term at index k=1 which is zero.
Its inverse binomial transform is 1, -1, 7/6, -3/2, 59/30, -5/2, 127/42, -7/2, 119/30, -9/2, 335/66, -11/2, ...; the numerators define this sequence here.

			The first few of the polynomials mentioned in the formula section are: 1, 1/2, 1/6 + x^2, (3/2)*x^2, -1/30 + x^2 + x^4, (5/2)*x^4, 1/42 - (1/2)*x^2 +(5/2)*x^4 + x^6, (7/2)*x^6, -1/30 + (2/3)*x^2 - (7/3)*x^4 + (14/3)*x^6 + x^8, (9/2)*x^8, ...  The values of these polynomials at x=1 start 1, 1/2, 7/6, 3/2, 59/30, 5/2, 127/42, 7/2, ... - _Peter Luschny_, Aug 18 2018

Cf. A176591 (denominators), A141056 (denominators for the unsigned variant).

read("transforms") ; evb := [1,0,seq(bernoulli(n),n=2..50)] ; BINOMIALi(evb) ; apply(numer,%) ; # R. J. Mathar, Dec 01 2010
seq(numer((-1)^n*(bernoulli(n,1)+bernoulli(n,2))/2),n=0..34); # Peter Luschny, Jun 17 2012
gf := cosh(x*z)*z/(1-exp(-z)): ser := series(gf, z, 35):
seq((-1)^n*numer(subs(x=1, n!*coeff(ser, z, n))), n=0..34); # Peter Luschny, Aug 19 2018

terms = 35; egf = (x/2)*((Exp[-x] + 1)/(Exp[x] - 1)) + O[x]^(terms);
CoefficientList[egf, x]*Range[0, terms-1]! // Numerator (* Jean-François Alcover, Jun 13 2017 *)

my(x = 'x + O('x^50)); apply(x->numerator(x), Vec(serlaplace((x/2)*(exp(-x) + 1)/(exp(x) - 1)))) \\ Michel Marcus, Aug 19 2018

Conjecture: a(2*n+1) = -2*n-1.
a(n) = numerator((-1)^n*(bernoulli(n, 1) + bernoulli(n, 2))/2). - Peter Luschny, Jun 17 2012
(-1)^n*a(n) are the numerators of the polynomials generated by cosh(x*z)*z/(1-exp(-z)) evaluated x=1 (see the example section). The denominators of these values are A141056. - Peter Luschny, Aug 18 2018

Apparently incorrect claims concerning the inverse binomial transform of the B_n removed by R. J. Mathar, Dec 01 2010

New name from Peter Luschny, Jun 17 2012

A176289 Denominators of the rational sequence with e.g.f. (x/2)*(1+exp(-x))/(1-exp(-x)).

Original entry on oeis.org

1, 1, 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, 6, 1, 510

Offset: 0

Paul Curtz, Apr 14 2010

Denominator of the Bernoulli number B_n, except a(1)=1. A minor variant of the Bernoulli denominators A027642.
The sequence of fractions A164555(n)/A027642(n) = 1/1, 1/2, 1/6, 0/1, -1/30, ...
and the sequence of fractions A027641(n)/A027642(n) = B_n = 1/1, -1/2, 1/6, 0/1, -1/30, ... differ only (by a sign) at n=1. The arithmetic mean of both sequences is 1/1, 0/1, 1/6, 0/1, -1/30, ..., equal to the aerated sequence A000367(n)/A002445(n). The definition here provides the denominators of this sequence of arithmetic means.

Antti Karttunen, Table of n, a(n) for n = 0..4096

Cf. A027641, A027642, A164555, A176327 (numerators), A141056.

seq(denom((bernoulli(i,0)+bernoulli(i,1))/2),i=0..64); # Peter Luschny, Jun 17 2012

Join[{1,1},Rest[Denominator[BernoulliB[Range[80]]]]] (* Harvey P. Dale, Jun 18 2012 *)

apply(deniominator, Vec(serlaplace((x/2)*(1+exp(-x))/(1-exp(-x))))) \\ Charles R Greathouse IV, Sep 26 2017

A176289(n) = if(1==n,n,denominator(bernfrac(n))); \\ Antti Karttunen, Dec 19 2018

a(2*n) = A002445(n), a(2*n+1)=1.

a(n) = A027642(n) for n <> 1.

More terms from Harvey P. Dale, May 03 2012

New name from Peter Luschny, Jun 18 2012

cp's OEIS Frontend

A176591 Bernoulli denominators A141056(n), with the exception a(1)=1.

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A213621 The denominator of the Bernoulli polynomial B(n,x) divided by the Clausen number C(n), A144845(n)/A141056(n).

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A213623 Numbers n such that the denominator of the Bernoulli polynomial B(n,x) equals the Clausen number C(n), {n | A144845(n) = A141056(n)}.

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A027642 Denominator of Bernoulli number B_n.

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A160014 Generalized Clausen numbers (table read by antidiagonals).

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A027760 Denominator of Sum_{p prime, p-1 divides n} 1/p.

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