A001067 -id:A001067 - cp's OEIS frontend

A300711 a(n) = A000367(n)/A001067(n).

1, 1, 1, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 7, 5, 1, 17, 1, 19, 1, 1, 11, 23, 1, 25, 13, 1, 7, 29, 1, 31, 1, 11, 17, 35, 1, 37, 19, 13, 1, 41, 1, 43, 11, 5, 23, 47, 1, 49, 1, 17, 13, 53, 1, 5, 7, 19, 29, 59, 1, 61, 31, 1, 1, 65, 11, 67, 17, 23, 7, 71, 1, 73, 37

Offset: 1

Bernd C. Kellner, Mar 11 2018

nonn

a(n) is the trivial factor of the numerator of Bernoulli(2n) that divides 2n.
The remaining part of the (unsigned) numerator equals a product of powers of irregular primes, or 1 if and only if n = 1, 2, 3, 4, 5, 7.
Alternatively, a(n) is the product over all prime powers p^e, where p^e is the highest power of p dividing 2n and p-1 does not divide 2n.

			a(5) = 5, since Bernoulli(10) = 5/66 and Bernoulli(10)/10 = 1/132.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..10000
Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007) 405-441.

A111008 equals the first entries and slightly differs, see a(35).

Cf. A000367, A001067, A193267, A195989, A300330, A002445, A075180.

using Nemo
function A300711(n)
    b = bernoulli(n)
    div(numerator(b), numerator(b*QQ(1,n)))
end
[A300711(n) for n in 2:2:148] |> println # Peter Luschny, Mar 11 2018

A300711 := proc(n) local P, F, f, divides; divides := (a,b) -> is(irem(b,a) = 0):
P := 1; F := ifactors(2*n)[2]; for f in F do if not divides(f[1]-1, 2*n) then
P := P*f[1]^f[2] fi od; P end: seq(A300711(n), n=1..74); # Peter Luschny, Mar 12 2018

Table[Numerator[BernoulliB[n]]/Numerator[BernoulliB[n]/n], {n, 2, 100, 2}]

a(n) = gcd(numerator(bernfrac(2*n)), 2*n) \\ Jianing Song, Apr 05 2021

upto(N)=bernvec(N);forstep(n=2,2*N,2,print1(gcd(numerator(bernfrac(n)), n),", ")) \\ Jeppe Stig Nielsen, Jun 22 2023

a(n) = numerator(Bernoulli(2n))/numerator(Bernoulli(2n)/(2n)).
a(n) * A195989(n) = n. - Peter Luschny, Mar 12 2018
From Jianing Song, Apr 05 2021: (Start)
a(n) = gcd(numerator(Bernoulli(2n)), 2n).
a(n) = A002445(n)*(2n)/A075180(2n-1). (End)

A033563 Primes in A001067.

Original entry on oeis.org

691, 3617, 43867, 657931, 151628697551, 26315271553053477373, 154210205991661, 1520097643918070802691, 923038305114085622008920911661422572613197507651

Offset: 1

Jud McCranie

nonn

All primes in this sequence are irregular primes. The next term, corresponding to Bernoulli(114), is too large to include. See A112548 for indices 2n that yield primes. - T. D. Noe, Sep 28 2005

Select[Table[Abs[Numerator[BernoulliB[2n]/2/n]], {n, 100}], PrimeQ] (Noe)

More terms from T. D. Noe, Sep 28 2005

A281331 Smallest prime factor of |A001067(n)|, or 1 if |A001067(n)| = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 691, 1, 3617, 43867, 283, 131, 103, 657931, 9349, 1721, 37, 151628697551, 26315271553053477373, 154210205991661, 137616929, 1520097643918070802691, 59, 383799511, 653, 417202699, 577, 39409, 113161, 67, 2003, 157, 1226592271, 839, 37, 688531, 3112655297839

Offset: 1

Seiichi Manyama, Jan 20 2017

nonn

			|A001067(10)| = 174611 = 283*617. So a(10) = 283.
|A001067(16)| = 7709321041217 = 37*683*305065927. So a(16) = 37.

Wikipedia, Kummer's congruences

Cf. A000928, A001067, A020639, A033563, A112548, A281332.

a[n_] := FactorInteger[Abs[Numerator[BernoulliB[2*n] / (2*n)]]][[1, 1]]; Table[a[n], {n, 1, 36}] (* Indranil Ghosh, Mar 12 2017 *)

a(n) = my(num = abs(numerator(bernfrac(2*n)/(2*n)))); if (num==1, 1, factor(num)[1,1]); \\ Michel Marcus, Jan 21 2017

a(n) = A020639(|A001067(n)|).
If n = A112548(m)/2, a(n) = |A001067(n)|.
a(18*m-2) = 37 for m > 0.

More terms from Michel Marcus, Jan 21 2017

A281332 Greatest prime factor of |A001067(n)|, or 1 if |A001067(n)| = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 691, 1, 3617, 43867, 617, 593, 2294797, 657931, 362903, 1001259881, 305065927, 151628697551, 26315271553053477373, 154210205991661, 1897170067619, 1520097643918070802691, 1798482437, 67568238839737, 153289748932447906241, 47464429777438199

Offset: 1

Seiichi Manyama, Jan 20 2017

nonn

			|A001067(10)| = 174611 = 283*617. So a(10) = 617.
|A001067(16)| = 7709321041217 = 37*683*305065927. So a(16) = 305065927.

Cf. A000928, A001067, A006530, A033563, A112548, A281331.

Table[FactorInteger[Abs@ Numerator[BernoulliB[2 n]/(2 n)]][[-1, 1]], {n, 25}] (* Michael De Vlieger, Jan 21 2017 *)

a(n) = if(abs(numerator(bernfrac(2*n) / (2*n))) == 1, 1, vecmax(factor(abs(numerator(bernfrac(2*n) / (2*n))))[,1]));
for(n=1, 25, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 11 2017

a(n) = A006530(|A001067(n)|).

If n = A112548(m)/2, a(n) = |A001067(n)|.

a(20)-a(25) from Michael De Vlieger, Jan 21 2017

A060309 A001067 appears to count the periodic points for a certain map. If so, then this is the sequence of the numbers of orbits of length n.

Original entry on oeis.org

1, 0, 0, 0, 0, 115, 0, 452, 4874, 17461, 7062, 19696950, 50610, 242341439, 114877883680, 481832564850, 8919335150, 1461959530725195586, 8116326631140, 13054135924822447372, 72385602091336704890, 115013510658268698717, 1127506827209663824722

Offset: 1

Thomas Ward, Apr 10 2001

nonn

			a(11) = 7062 because the 11th term of A001067 is 77683 and the first term is 1, so there should be (77683-1)/11 = 7062 orbits of length 11.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
T. Ward, Exactly realizable sequences

Cf. A001067, A060171, A060479.

If b(n) is the n-th term of A001067, then a(n)=(1/n)* |Sum_{d|n}mu(d)b(n/d)|, n<>2.

a(18) corrected and more terms from Sean A. Irvine, Nov 08 2022

A006953 a(n) = denominator of Bernoulli(2n)/(2n).

Original entry on oeis.org

12, 120, 252, 240, 132, 32760, 12, 8160, 14364, 6600, 276, 65520, 12, 3480, 85932, 16320, 12, 69090840, 12, 541200, 75852, 2760, 564, 2227680, 132, 6360, 43092, 6960, 708, 3407203800, 12, 32640, 388332, 120, 9372, 10087262640

Offset: 1

Simon Plouffe and N. J. A. Sloane

a(n) are alternately divisible by 12 and 120, a(n)/(12, 120, 12, 120, 12, 120, ...) = 1, 1, 21, 2, 11, 273, ... . - Paul Curtz, Sep 13 2011 and Michel Marcus, Jan 05 2013
A141590/(2 before a(n+1)) = 1/2 + 1/12 - 1/120 + 1/252 is an old semi-convergent series for Euler's constants A001620 ("2 before a" meaning that one term, namely 2, is inserted before the sequence). This series is discussed in details in reference [Blagouchine, 2016], Sect. 3 and Fig. 3. - Paul Curtz, Sep 13 2011, Michel Marcus, Jan 05 2013 and Iaroslav V. Blagouchine, Sep 16 2015
a(n) = A006863(n)/2. - Michel Marcus, Jan 05 2013

			Sequence Bernoulli(2n)/(2n) (n >= 1) begins 1/12, -1/120, 1/252, -1/240, 1/132, -691/32760, 1/12, -3617/8160, ... .

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 259, (6.3.18) and (6.3.19); also p. 810.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..1000
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].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 259, (6.3.18) and (6.3.19).
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only, Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.
R. D. Carmichael, Notes on the simplex theory of numbers, Bull. Amer. Math. Soc. 15 (1909), 217-223.
G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, Integer Sequences and Periodic Points, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3
E. Z. Goren, Tables of values of Riemann zeta functions
A. Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function
J. Sondow and E. W. Weisstein, MathWorld: Harmonic Number
Index entries for sequences related to Bernoulli numbers.

Numerators are given by A001067.

List([1..40], n-> DenominatorRat(Bernoulli(2*n)/(2*n)) ); # G. C. Greubel, Sep 19 2019

[Denominator(Bernoulli(2*n)/(2*n)):n in [1..40]]; // Vincenzo Librandi, Sep 17 2015

A006953_list := proc(n) 1/(1-1/exp(z)); series(%,z,2*n+4);
seq(denom((-1)^i*(2*i+1)!*coeff(%,z,2*i+1)),i=0..n) end;
A006953_list(35); # Peter Luschny, Jul 12 2012

Table[Denominator[BernoulliB[2n]/(2n)],{n,40}] (* Harvey P. Dale, Jan 12 2022 *)

a(n) = denominator(bernfrac(2*n)/(2*n)); \\ Michel Marcus, Apr 21 2016

[denominator(bernoulli(2*n)/(2*n)) for n in (1..40)] # G. C. Greubel, Sep 19 2019

Zeta(1-2*n) = -Bernoulli(2*n)/(2*n).
G.f. for Bernoulli(2*n)/(2*n) = A001067(n)/A006953(n): (-1)^n/((2*Pi)^(2*n)*(2*n)) * Integral_{t=0..1} log(1-1/t)^(2*n) dt. - Gerry Martens, May 18 2011
E.g.f.: a(n) = denominator((2*n+1)!*[x^(2*n+1)](1/(1-1/exp(x)))). - Peter Luschny, Jul 12 2012

Previous Mathematica program replaced by Harvey P. Dale, Jan 12 2022

A006863 Denominator of B_{2n}/(-4n), where B_m are the Bernoulli numbers.

Original entry on oeis.org

1, 24, 240, 504, 480, 264, 65520, 24, 16320, 28728, 13200, 552, 131040, 24, 6960, 171864, 32640, 24, 138181680, 24, 1082400, 151704, 5520, 1128, 4455360, 264, 12720, 86184, 13920, 1416, 6814407600, 24

Offset: 0

N. J. A. Sloane, Jeffrey Shallit, Simon Plouffe

Carmichael defines lambda(n) to be the exponent of the group U(n) of units of the integers mod n. He shows that given m there is a number lambda^*(m) such that lambda(n) divides m if and only if n divides lambda^*(m). He gives a formula for lambda^*(m), equivalent to the one I've quoted for even m. (We have lambda^*(m)=2 for any odd m.) The present sequence gives the values of lambda^*(2m) for positive integers m. - Peter J. Cameron, Mar 25 2002
(-1)^n*B_{2n}/(-4n) = Integral_{t>=0} t^(2n-1)/(exp(2*Pi*t) - 1)dt. - Benoit Cloitre, Apr 04 2002
Michael Lugo (see link) conjectures, and Peter McNamara proves, that a(n) = gcd_{ primes p > 2n+1 } (p^(2n) - 1). - Tanya Khovanova, Feb 21 2009 [edited by Charles R Greathouse IV, Dec 03 2014]

Bruce Berndt, Ramanujan's Notebooks Part II, Springer-Verlag; see Integrals and Asymptotic Expansions, p. 220.
F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg, 2nd ed. 1994, p. 130.
J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 286.
Douglas C. Ravenel, Complex cobordism theory for number theorists, Lecture Notes in Mathematics, 1326, Springer-Verlag, Berlin-New York, 1988, pp. 123-133.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. C. Vaughan and T. D. Wooley, Waring's problem: a survey, pp. 285-324 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003. (The function K(2n), see p. 303.)

T. D. Noe, Table of n, a(n) for n = 0..10000
P. J. Cameron and D. A. Preece, Notes on primitive lambda-roots
R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1909-10), 232-238.
G. Everest, Y. Puri and T. Ward, Integer sequences counting periodic points, arXiv:math/0204173 [math.NT], 2002.
Michael Lugo, A little number theory problem (2008)
Eric Weisstein's World of Mathematics, Eisenstein Series.
Index entries for sequences related to Bernoulli numbers.

Numerators are A001067.

Cf. A000367/A002445, A002322, A079612.

Concatenation([1], List([1..35], n-> DenominatorRat(Bernoulli(2*n)/(-4*n)) )); # G. C. Greubel, Sep 19 2019

[1] cat [Denominator(Bernoulli(2*n)/(-4*n)):n in [1..35]]; // G. C. Greubel, Sep 19 2019

1,seq(denom(bernoulli(2*n)/(-4*n)), n=1 .. 100); # Robert Israel, Dec 03 2014

a[n_] := Denominator[BernoulliB[2n]/(-4n)]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Mar 20 2011 *)

a(n) = if (n == 0, 1, denominator(bernfrac(2*n)/(-4*n))); \\ Michel Marcus, Sep 10 2013

[1]+[denominator(bernoulli(2*n)/(-4*n)) for n in (1..35)] # G. C. Greubel, Sep 19 2019

B_{2k}/(4k) = -(1/2)*zeta(1-2k). For n > 0, a(n) = gcd k^L (k^{2n}-1) where k ranges over all the integers and L is as large as necessary.

Product of 2^{a+2} (where 2^a exactly divides 2*n) and p^{a+1} (where p is an odd prime such that p-1 divides 2*n and p^a exactly divides 2*n). - Peter J. Cameron, Mar 25 2002

Thanks to Michael Somos for helpful comments.

A262235 Denominators of a series leading to Euler's constant gamma.

Original entry on oeis.org

4, 72, 32, 14400, 1728, 2540160, 138240, 261273600, 896000, 10538035200, 209018880, 407994402816000, 5633058816000, 941525544960000, 4723310592, 8707228239790080000, 6162712657920000, 17473102222724628480000, 107559878256230400000, 14162409169997856768000000

Offset: 1

Iaroslav V. Blagouchine, Sep 15 2015

Gamma = 1 - 1/4 - 5/72 - 1/32 - 251/14400 - 19/1728 - 19087/2540160 - ..., see the references below.

			Denominators of 1/4, 5/72, 1/32, 251/14400, 19/1728, 19087/2540160, ...

G. C. Greubel, Table of n, a(n) for n = 1..250
Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi, Journal of Mathematical Analysis and Applications (Elsevier), 2016. arXiv version, arXiv:1408.3902 [math.NT], 2014-2016.
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only. Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.

Cf. A001067, A001620, A002657, A002790, A006953, A075266, A075267, A195189.

a := proc(n) local r; r := proc(n) option remember; if n=0 then 1 else
1 - add(r(k)/(n-k+1), k=0..n-1) fi end: denom(r(n)/(n*(n+1))) end:
seq(a(n), n=1..20); # Peter Luschny, Apr 19 2018

g[n_] := Sum[Abs[StirlingS1[n, l]]/(l + 1), {l, 1, n}]/(n*(n + 1)!); a[n_] := Denominator[g[n]]; Table[a[n], {n, 1, 20}]

a(n) = C2(n)/(n*(n + 1)!), where C2(n) are Cauchy numbers of the second kind (see A002657 and A002790).

A090495 Numbers k such that numerator(Bernoulli(2k)/(2k)) is different from numerator(Bernoulli(2k)/(2k(2k-1))).

Original entry on oeis.org

574, 1185, 1240, 1269, 1376, 1906, 1910, 2572, 2689, 2980, 3238, 3384, 3801, 3904, 4121, 4570, 4691, 4789, 5236, 5862, 5902, 6227, 6332, 6402, 6438, 6568, 7234, 7900, 8113, 8434, 8543, 8557, 8566, 9232, 9611, 9670, 9824, 9891, 9898, 10564, 10587, 10754, 11230, 11247, 11535, 11691, 11896, 12562, 12965, 13019, 13228, 13246, 13355, 13484, 13894, 14560, 14714, 14957, 15176, 15226, 15346, 15892, 16558, 16668, 16944, 17035, 17224, 17387, 17890, 18379, 18406, 18534, 18556, 18761, 19222, 19598, 19888, 20090

Offset: 1

N. J. A. Sloane, Feb 03 2004

Michael Somos (Feb 01 2004) discovered the remarkable fact that A001067 is different from A046968, even though they agree for the first 573 terms.
Numbers n such that A001067 is different from A046968, or alternatively, those n such that gcd(A001067(n),2n-1) is > 1.
If gcd(A000367(n), A000367(n+2)) <>1 then n = A090495(n) - (3*A090496(n) + 1)/2. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 08 2004
So far, all terms correspond to irregular primes. Notice that these numbers are generated by n=((2k+1)p+1)/2 where p is an irregular prime and k is some integer = 1,2,... . In the Excel spreadsheet provided at the link, you will notice that much larger firstborn irregular primes p tend to produce smaller values of k. E.g., p = 691, 683, 653, k = 5, 15, 23. So by some guessing we could test a given large irregular prime for the first few values of k. I found ip's 257, 293, 311 this way, but not the index. Also the spreadsheet shows the corresponding irregular primes where the Bacher forecast fails for firstborn irregular prime. - Cino Hilliard, Feb 15 2004

Robert G. Wilson v, Table of n, a(n) for n = 1..200
Cino Hilliard, Bernoulli ratios [posted on Yahoo group B2LCC, Feb 04 2004]
Eric Weisstein's World of Mathematics, Stirling's Series

Cf. A090496, A001067, A046968, A092291. A274297.

a := n->numer(bernoulli(2*n)/(2*n)): b := n->numer(bernoulli(2*n)/(2*n*(2*n-1))): for n from 1 to 2000 do if a(n)<>b(n) then print(n,a(n)/b(n)); fi; od:

a[n_] := Numerator[BernoulliB[2n]/(2n)] (* A001067 *); b[n_] := Numerator[BernoulliB[2n]/(2n(2n-1))] (* A046968 *); For[n=1, n <= 580, n++, If[ a[n] != b[n], Print[n, " ", a[n]/b[n]] ] ]
k = 1; lst = {}; While[k < 38001, b = BernoulliB[2 k]; If [Numerator[b/(2 k)] != Numerator[b/(2 k (2 k - 1))], AppendTo[lst, k]; Print[{k}]]; k++ ]; lst (* Robert G. Wilson v, Aug 19 2010 *)

bern2(c,m1,m2) = { for(n=m1,m2, n2=n+n; a = numerator(bernfrac(n2)/(n2)); \ A001067 b = numerator(a/(n2-1)); if(a <> b,print("A("c") = "n","a/b);c++) ) } \\ Cino Hilliard

a(1)-a(7) from Michael Somos and W. Edwin Clark, Feb 03 2004
a(8)-a(9) from Robert G. Wilson v, Feb 03 2004
a(10)-a(12) from Eric W. Weisstein, Feb 03 2004
a(13)-a(39) from Cino Hilliard, Feb 03 2004
a(40) from Eric W. Weisstein, Feb 04 2004
Many further terms from Cino Hilliard, Feb 15 2004

A046968 Numerators of coefficients in Stirling's expansion for log(Gamma(z)).

Original entry on oeis.org

1, -1, 1, -1, 1, -691, 1, -3617, 43867, -174611, 77683, -236364091, 657931, -3392780147, 1723168255201, -7709321041217, 151628697551, -26315271553053477373, 154210205991661, -261082718496449122051, 1520097643918070802691

Offset: 1

Douglas Stoll (dougstoll(AT)email.msn.com)

A001067(n) = a(n) if n<574; A001067(574) = 37*a(574). - Michael Somos, Feb 01 2004

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 257, Eq. 6.1.41.
L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979, p. 205

Seiichi Manyama, Table of n, a(n) for n = 1..314
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].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 257, Eq. 6.1.41.
R. P. Brent, Asymptotic approximation of central binomial coefficients with rigorous error bounds, arXiv:1608.04834 [math.NA], 2016.
N. Elezovic, Asymptotic Expansions of Central Binomial Coefficients and Catalan Numbers, J. Int. Seq. 17 (2014) # 14.2.1.
Eric Weisstein's World of Mathematics, Stirling's Series
Index entries for sequences related to Bernoulli numbers.

Denominators given by A046969.

Similar to but different from A001067. See A090495, A090496.

List([1..25], n-> NumeratorRat(Bernoulli(2*n)/(2*n*(2*n-1))) ); # G. C. Greubel, Sep 19 2019

[Numerator(Bernoulli(2*n)/(2*n*(2*n-1))): n in [1..25]]; // G. C. Greubel, Sep 19 2019

seq(numer(bernoulli(2*n)/(2*n*(2*n-1))), n = 1..25); # G. C. Greubel, Sep 19 2019

Table[ Numerator[ BernoulliB[2n]/(2n(2n - 1))], {n, 1, 22}] (* Robert G. Wilson v, Feb 03 2004 *)
s = LogGamma[z] + z - (z - 1/2) Log[z] - Log[2 Pi]/2 + O[z, Infinity]^42; DeleteCases[CoefficientList[s, 1/z], 0] // Numerator (* Jean-François Alcover, Jun 13 2017 *)

a(n)=if(n<1,0,numerator(bernfrac(2*n)/(2*n)/(2*n-1)))

[numerator(bernoulli(2*n)/(2*n*(2*n-1))) for n in (1..25)] # G. C. Greubel, Sep 19 2019

From numerator of Jk(z) = (-1)^(k-1)*Bk/(((2k)*(2k-1))*z^(2k-1)), so Gamma(z) = sqrt(2*Pi)*z^(z-0.5)*exp(-z)*exp(J(z)).

More terms from Frank Ellermann, Jun 13 2001

cp's OEIS Frontend

A300711 a(n) = A000367(n)/A001067(n).

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A033563 Primes in A001067.

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A281331 Smallest prime factor of |A001067(n)|, or 1 if |A001067(n)| = 1.

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A281332 Greatest prime factor of |A001067(n)|, or 1 if |A001067(n)| = 1.

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A060309 A001067 appears to count the periodic points for a certain map. If so, then this is the sequence of the numbers of orbits of length n.

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A006953 a(n) = denominator of Bernoulli(2n)/(2n).

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A006863 Denominator of B_{2n}/(-4n), where B_m are the Bernoulli numbers.

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A090495 Numbers k such that numerator(Bernoulli(2k)/(2k)) is different from numerator(Bernoulli(2k)/(2k(2k-1))).