A001067 -id:A001067 - cp's OEIS frontend

A060054 Numerators of numbers appearing in the Euler-Maclaurin summation formula.

-1, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -691, 0, 1, 0, -3617, 0, 43867, 0, -174611, 0, 77683, 0, -236364091, 0, 657931, 0, -3392780147, 0, 1723168255201, 0, -7709321041217, 0, 151628697551, 0, -26315271553053477373

Offset: 1

Wolfdieter Lang, Feb 16 2001

a(n+1) = numerator(-Zeta(-n)), n>=1, with Riemann's zeta function. a(1)=-1=-numerator(-Zeta(-0)). For denominators see A075180.
Comment from N. J. A. Sloane, Oct 15 2008: (Start)
It appears that essentially the same sequence of rational numbers arises when we expand 1/(exp(1/x)-1) for large x. Here is the result of applying Bruno Salvy's gdev Maple program (answering a question raised by Roger L. Bagula):
gdev(1/(exp(1/x)-1), x=infinity, 20);
x - 1/2 + (1/12)/x - (1/720)/x^3 + (1/30240)/x^5 - (1/1209600)/x^7 + (1/47900160)/x^9 - (691/1307674368000)/x^11 + (1/74724249600)/x^13 - (3617/10670622842880000)/x^15 + (43867/5109094217170944000)/x^17 - (174611/802857662698291200000)/x^19 + ... (End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 16 (3.6.28), p. 806 (23.1.30), p. 886 (25.4.7).

Vincenzo Librandi, Table of n, a(n) for n = 1..600
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. 16 (3.6.28), p. 806 (23.1.30), p. 886 (25.4.7).
Zhanna Kuznetsova and Francesco Toppan, Classification of minimal Z_2 X Z_2-graded Lie (super)algebras and some applications, arXiv:2103.04385 [math-ph], 2021.

Denominators of nonzero numbers give A060055.
Cf. A001067 (numerator of B(2*k)/(2*k)).
Cf. A075180.
Cf. also A120082/A227830.
Cf. A242179, A106831, A000142.

a060054 n = a060054_list !! n
a060054_list = -1 : map (numerator . sum) (tail $ zipWith (zipWith (%))
   (zipWith (map . (*)) a000142_list a242179_tabf) a106831_tabf)
-- Reinhard Zumkeller, Jul 04 2014

a[m_] := Sum[(-2)^(-k-1) k! StirlingS2[m,k],{k,0,m}]/(2^(m+1)-1); Table[Numerator[a[i]], {i,0,30}] (* Peter Luschny, Apr 29 2009 *)

a(n):=num((-1)^n*sum(binomial(n+k-1,n-1)*sum((j!*(-1)^(j)*binomial(k,j)*stirling1(n+j,j))/(n+j)!,j,1,k),k,1,n)); /* Vladimir Kruchinin, Feb 03 2013 */

a(n) = numerator(b(n)) with b(1) = -1/2; b(2*k+1) = 0, k >= 1; b(2*k) = B(2*k)/(2*k)! (B(2*n) = B_{2n} Bernoulli numbers: numerators A000367, denominators A002445)

A090496 Ratio of numerator(Bernoulli(2n)/(2n)) to numerator(Bernoulli(2n)/(2n(2n-1))) for n's for which they are different.

Original entry on oeis.org

37, 103, 37, 59, 131, 37, 67, 37, 283, 59, 37, 101, 691, 37, 67, 37, 59, 157, 37, 617, 37, 593, 67, 59, 103, 37, 37, 37, 59, 101, 67, 157, 37, 37, 149, 233, 59, 131, 37, 37, 683, 67, 37, 271, 59, 103, 37, 37, 67, 263, 37, 59, 307, 101, 37, 37, 577, 59, 67, 37, 653, 37, 37, 59, 103, 157, 37, 67, 37, 59, 131, 101

Offset: 1

N. J. A. Sloane, Feb 03 2004

A001067(n) / A046968(n) when they are different, or alternatively, gcd(A001067(n),2n-1) when that number is > 1.
These numbers are always products of irregular primes (A000928).
All values yielding 37 are of the form 574+666*k, k=0,1,2,3,4,... and form thus an arithmetic progression with step 666=18*37=((37-1)/2)*37. All values yielding 59 are of the form 1269+1711*k, k=0,1,2,3 and 1711=28*59=((59-1)/2)*59. The two values yielding 67 are at distance 2211=((67-1)/2)*67. Conjecture: all indices yielding a given prime p form an arithmetic progression of step ((p-1)/2)*p. See A092291. - Roland Bacher, Feb 04 2004
The positions where 37 occurs appear to coincide with A026352. - Mohammed Bouayoun, Feb 05 2004
Roland Bacher conjectures that values of n yielding the same quotient p form an arithmetic progression n0+d*k, where d = p(p-1)/2. Actual and conjectured values of n0 are in the sequence A092291.
Composite values do occur. An example is 2n = 272876, which yields a quotient of 37*59. This was found by tdn using the Kummer congruences and CRT: using the irregular pairs (37,32) and (59,44), we know that the following Diophantine equations must be solved for (k,l,m): 32+36*k = 44+58*l = 1+37*59*m. Some quotients are not possible, e.g., 37*67, 37*103. All quotients are the product of irregular primes A000928. Composite quotients imply there are missing terms in the arithmetic progression conjectured by Bacher. - T. D. Noe, Feb 12 2004

Amiram Eldar, Table of n, a(n) for n = 1..200 (calculated from the b-file at A090495)
Bernd Kellner, A conjecture about numerators of Bernoulli numbers
Eric Weisstein's World of Mathematics, Stirling's Series

Cf. A090495, A001067, A046968, A092291.

A090496 = {}; Do[ r = Numerator[ b = BernoulliB[2n]/(2n) ] / Numerator[ b/(2n-1) ]; If[ r > 1, Print[n, " ", r]; AppendTo[ A090496, r] ], {n, 1, 20000}]; A090496 (* Jean-François Alcover, Jan 24 2012 *)

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)-a(44) from Eric W. Weisstein, Feb 04 2004
Terms from a(45) onwards from David Wasserman, Dec 06 2005

A141590 a(n) = numerator of Bernoulli(2n)/(2n + 1)!. Bisection of A120082.

Original entry on oeis.org

1, 1, -1, 1, -1, 1, -691, 1, -3617, 43867, -174611, 77683, -236364091, 657931, -3392780147, 1723168255201, -7709321041217, 151628697551, -26315271553053477373, 154210205991661, -261082718496449122051, 1520097643918070802691, -2530297234481911294093

Offset: 0

Paul Curtz, Aug 20 2008

Numerators of the Taylor expansion coefficients of the Debye function D(1,x) at the even powers of x.

			Note that a(34) = -125235502160125163977598011460214000388469 but A255505(34) = -4633713579924631067171126424027918014373353.

Peter Luschny, Table of n, a(n) for n = 0..300
Kevin Acres and David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018.

Cf. A000367, A001067, A046968, A120082, A255505.

A141590:= func< n | Numerator(BernoulliNumber(2*n)/Factorial(2*n+1)) >;
[A141590(n): n in [0..35]]; // G. C. Greubel, Sep 16 2024

A141590 := proc(n) A120082(2*n) end:
seq(A141590(n), n=0..30) ; # R. J. Mathar, Sep 03 2009
seq(numer(bernoulli(2*n)/(2*n+1)!), n=0..34); # Peter Luschny, Dec 03 2022

Table[Numerator[BernoulliB[2*n]/(2*n+1)!], {n,0,35}] (* G. C. Greubel, Sep 16 2024 *)

def A141590(n): return numerator(bernoulli(2*n)/factorial(2*n+1))
[A141590(n) for n in range(36)] # G. C. Greubel, Sep 16 2024

a(n) = A120082(2*n).

Edited and extended by R. J. Mathar, Sep 03 2009

Edited by Peter Luschny, Dec 03 2022

A112548 Numbers k such that the numerator of Bernoulli(k)/k is (apart from sign) prime.

Original entry on oeis.org

12, 16, 18, 26, 34, 36, 38, 42, 74, 114, 118, 396, 674, 1870, 4306, 22808

Offset: 1

T. D. Noe, Sep 28 2005

In 1911 Ramanujan believed that the numerator of Bernoulli(k)/k for k even was (apart from sign) always either 1 or a prime. This is false.
Equivalently, k such that the numerator of zeta(1-k) is prime. No other k < 23000. Kellner's Calcbn program was used to generate the numerators of Bernoulli(k)/k for k > 5000. Mathematica and PFGW were used to test for probable primes. David Broadhurst found n=4306, which yields a 10342-digit probable prime. For n<4306, the primes have been proved. Bouk de Water proved the prime for n=1870. All these primes are necessarily irregular.
The number generated by k=4306 was recently proved prime. See Chris Caldwell's link for more details. - T. D. Noe, Apr 06 2009
a(17) > 50000. - Robert Price, Oct 20 2013
a(17) > 74708. - Simon Plouffe, Mar 06 2022
a(17) > 270000. - Serge Batalov, Jun 26 2025

S. Ramanujan, Some properties of Bernoulli's numbers, J. Indian Math. Soc., 3 (1911), 219-234.

Bernd Kellner, Program Calcbn - A program for calculating Bernoulli numbers
Chris Caldwell, Top twenty irregular primes
K. Ono, Honoring a gift from Kumbakonam, Notices Amer. Math. Soc., 53 (2006), 640-651.
Simon Plouffe, Primes as sums of irrational numbers, Preprint, 2016.
Eric Weisstein's World of Mathematics, Irregular Prime

Cf. A001067 (numerator of Bernoulli(2n)/(2n)).
Cf. A033563 (primes in A001067).
Cf. A092132 (n such that the numerator of Bernoulli(n) is prime).
Cf. A112741 (primes p such that zeta(1-2p)/zeta(-1) is prime).
Cf. A119766.

A112548 := proc(nmax) local numr; for n from 2 to nmax by 2 do numr := abs(numer(bernoulli(n)/n)) ; if isprime(numr) then print(n) ; fi ; od ; end : A112548(3000) ; # R. J. Mathar, Jun 21 2006

Select[Range[2, 5000, 2], PrimeQ[Numerator[BernoulliB[ # ]/# ]]&]

A262382 Numerators of a semi-convergent series leading to the first Stieltjes constant gamma_1.

Original entry on oeis.org

-1, 11, -137, 121, -7129, 57844301, -1145993, 4325053069, -1848652896341, 48069674759189, -1464950131199, 105020512675255609, -22404210159235777, 1060366791013567384441, -15899753637685210768473787, 2241672100026760127622163469, -8138835628210212414423299

Offset: 1

Iaroslav V. Blagouchine, Sep 20 2015

gamma_1 = - 1/12 + 11/720 - 137/15120 + 121/11200 - 7129/332640 + 57844301/908107200 - ..., see formulas (46)-(47) in the reference below.

			Numerators of -1/12, 11/720, -137/15120, 121/11200, -7129/332640, 57844301/908107200, ...

G. C. Greubel, Table of n, a(n) for n = 1..259
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. A001620, A002206, A195189, A075266, A262235, A001067, A006953, A082633, A262383 (denominators of this series), A086279, A086280, A262387.

a := n -> numer(Zeta(1 - 2*n)*(Psi(2*n) + gamma)):
seq(a(n), n=1..16); # Peter Luschny, Apr 19 2018

a[n_] := Numerator[-BernoulliB[2*n]*HarmonicNumber[2*n - 1]/(2*n)]; Table[a[n], {n, 1, 20}]

a(n) = numerator(-bernfrac(2*n)*sum(k=1,2*n-1,1/k)/(2*n)); \\ Michel Marcus, Sep 23 2015

a(n) = numerator(-B_{2n}*H_{2n-1}/(2n)), where B_n and H_n are Bernoulli and harmonic numbers respectively.

a(n) = numerator(Zeta(1 - 2*n)*(Psi(2*n) + gamma)), where gamma is Euler's gamma. - Peter Luschny, Apr 19 2018

A262383 Denominators of a semi-convergent series leading to the first Stieltjes constant gamma_1.

Original entry on oeis.org

12, 720, 15120, 11200, 332640, 908107200, 4324320, 2940537600, 175991175360, 512143632000, 1427794368, 7795757249280, 107084577600, 279490747536000, 200143324310529600, 1178332991611776000, 157531148611200, 906996615309386784000, 5828652498614400, 262872227687509440000

Offset: 1

Iaroslav V. Blagouchine, Sep 20 2015

gamma_1 = - 1/12 + 11/720 - 137/15120 + 121/11200 - 7129/332640 + 57844301/908107200 - ..., see formulas (46)-(47) in the reference below.

			Denominators of -1/12, 11/720, -137/15120, 121/11200, -7129/332640, 57844301/908107200, ...

G. C. Greubel, Table of n, a(n) for n = 1..500
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. A001620, A002206, A195189, A075266, A262235, A001067, A006953, A082633, A262382 (numerators of this series), A086279, A086280, A262387.

a := n -> denom(Zeta(1 - 2*n)*(Psi(2*n) + gamma)):
seq(a(n), n=1..20); # Peter Luschny, Apr 19 2018

a[n_] := Denominator[-BernoulliB[2*n]*HarmonicNumber[2*n - 1]/(2*n)]; Table[a[n], {n, 1, 20}]

a(n) = denominator(-bernfrac(2*n)*sum(k=1,2*n-1,1/k)/(2*n)); \\ Michel Marcus, Sep 23 2015

a(n) = denominator(-B_{2n}*H_{2n-1}/(2n)), where B_n and H_n are Bernoulli and harmonic numbers respectively.

a(n) = denominator(Zeta(1 - 2*n)*(Psi(2*n) + gamma)), where gamma is Euler's gamma. - Peter Luschny, Apr 19 2018

A262387 Denominators of a semi-convergent series leading to the third Stieltjes constant gamma_3.

Original entry on oeis.org

1, 120, 1008, 28800, 49896, 101088000, 5702400, 12350257920000, 43480172736000, 7075668600000, 206069667148800, 5919216795588096000, 581222138112000, 8460252005694128640000, 18991807088644406016000, 1150594272774401495040000, 33940540399314092544000, 9737059611553100811150566400000, 1290633707289706940160000, 1263402804161736165764268432000000

Offset: 1

Iaroslav V. Blagouchine, Sep 20 2015

gamma_3 = + 1/120 - 17/1008 + 967/28800 - 4523/49896 + 33735311/101088000 - ..., see formulas (46)-(47) in the reference below.

			Denominators of -0/1, 1/120, -17/1008, 967/28800, -4523/49896, 33735311/101088000, ...

G. C. Greubel, Table of n, a(n) for n = 1..500
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, 2015.

Cf. A001620, A002206, A195189, A075266, A262235, A001067, A006953, A082633, A262382, A262383, A086279, A262384, A262385, A086280, A262386 (numerators of this series).

a[n_] := Denominator[-BernoulliB[2*n]*(HarmonicNumber[2*n - 1]^3 - 3*HarmonicNumber[2*n - 1]*HarmonicNumber[2*n - 1, 2] + 2*HarmonicNumber[2*n - 1, 3])/(2*n)]; Table[a[n], {n, 1, 20}]

a(n) = denominator(-bernfrac(2*n)*(sum(k=1,2*n-1,1/k)^3 -3*sum(k=1,2*n-1,1/k)*sum(k=1,2*n-1,1/k^2) + 2*sum(k=1,2*n-1,1/k^3))/(2*n));

a(n) = denominator(-B_{2n}*(H^3_{2n-1}-3*H_{2n-1}*H^(2){2n-1}+2*H^(3){2n-1})/(2n)), where B_n, H_n and H^(k)_n are Bernoulli, harmonic and generalized harmonic numbers respectively.

A241601 Largest divisor of A246006(n) whose prime factors are all >= n+2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 61, 1, 277, 1, 50521, 691, 41581, 1, 199360981, 3617, 228135437, 43867, 2404879675441, 174611, 14814847529501, 77683, 69348874393137901, 236364091, 238685140977801337, 657931, 4087072509293123892361, 3392780147, 454540704683713199807

Offset: 0

Eric Chen, Dec 15 2014

nonn

Notice: Not all a(n) are 1 or primes, the first example is a(11) = 50521, it equals 19*2659.
a(2n) is a product of powers of Bernoulli irregular primes (A000928), with the exception of n = 0,1,2,3,4,5,7.
a(2n+1) is a product of powers of Euler irregular primes (A120337), with the exception of n = 0,1,2.
Conjectures: All terms are squarefree, and there are infinitely many n such that a(n) is prime.
a(n) = 1 iff n is in the set {0, 1, 2, 3, 4, 5, 6, 8, 10, 14}.
a(n) is prime for n = {7, 9, 12, 16, 17, 18, 26, 34, 36, 38, 39, 42, 49, 74, 114, 118, ...}.
All prime factors of a(n) are irregular primes (Bernoulli or Euler) and with an irregular pair to n: (61, 7), (277, 9), (19, 11), (2659, 11), (691, 12), (43, 13), (967, 13), (47, 15), (4241723, 15), (3617, 16), (228135437, 17), (43867, 18), (79, 19), (349, 19), (84224971, 19), ...
Number of ns such that a prime p divides a(n) is the irregular index of p, for example, 67 divides both a(27) and a(58), so it has irregular index two.
a(149) is the first a(n) which is not completely factored (with a 202-digit composite remaining).

Cf. A246006.
Cf. A000928, A120337.
Cf. A001067, A120082, A141590, A060054, A091912.

b[n_] := Numerator[BernoulliB[2 n]/(2 n)];
c[n_] := Numerator[SeriesCoefficient[Log[Tan[x]+1/Cos[x]], {x, 0, 2n+1}]];
a[0] = 1; a[n_] := If[EvenQ[n], b[n/2] // Abs, c[(n-1)/2]];
Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Jul 03 2019 *)

a(2n) = |A001067(n)| = |A120082(2n)| = |A141590(n)| = |A060054(n)|.

a(2n+1) = A091912(n).

A262384 Numerators of a semi-convergent series leading to the second Stieltjes constant gamma_2.

Original entry on oeis.org

0, -1, 5, -469, 6515, -131672123, 63427, -47800416479, 15112153995391, -29632323552377537, 4843119962464267, -1882558877249847563479, 2432942522372150087, -2768809380553055597986831, 334463513629004852735064113, -1125061940756859461946444233539, 333807583501528759350875247323

Offset: 1

Iaroslav V. Blagouchine, Sep 20 2015

gamma_2 = - 1/60 + 5/336 - 469/21600 + 6515/133056 - 131672123/825552000 + ..., see formulas (46)-(47) in the reference below.

			Numerators of 0/1, -1/60, 5/336, -469/21600, 6515/133056, -131672123/825552000, ...

G. C. Greubel, Table of n, a(n) for n = 1..237
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.

The sequence of denominators is A262385.

Cf. A001067, A001620, A002206, A006953, A075266, A082633, A086279, A086280, A195189, A262235, A262382, A262383, A262386, A262387.

a := n -> numer(-Zeta(1 - 2*n)*(Psi(1, 2*n) + (Psi(0,2*n) + gamma)^2 - (Pi^2)/6)):
seq(a(n), n=1..17); # Peter Luschny, Apr 19 2018

a[n_] := Numerator[BernoulliB[2*n]*(HarmonicNumber[2*n - 1]^2 - HarmonicNumber[2*n - 1, 2])/(2*n)]; Table[a[n], {n, 1, 20}]

a(n) = numerator(bernfrac(2*n)*(sum(k=1,2*n-1,1/k)^2 - sum(k=1,2*n-1,1/k^2))/(2*n)); \\ Michel Marcus, Sep 23 2015

a(n) = numerator(B_{2n}*(H^2_{2n-1}-H^(2)_{2n-1})/(2n)), where B_n, H_n and H^(k)_n are Bernoulli, harmonic and generalized harmonic numbers respectively.

a(n) = numerator(-Zeta(1 - 2*n)*(Psi(1,2*n) + (Psi(0,2*n) + gamma)^2 - (Pi^2)/6)), where gamma is Euler's gamma and Psi is the digamma function. - Peter Luschny, Apr 19 2018

A262385 Denominators of a semi-convergent series leading to the second Stieltjes constant gamma_2.

Original entry on oeis.org

1, 60, 336, 21600, 133056, 825552000, 89100, 11435424000, 483113030400, 101889627840000, 1471926193920, 42280119968486400, 3425059028160, 209827678712652000, 1184296360402995840, 163066081742403840000, 1749151741873536000, 20373357051590182072392960000

Offset: 1

Iaroslav V. Blagouchine, Sep 20 2015

gamma_2 = - 1/60 + 5/336 - 469/21600 + 6515/133056 - 131672123/825552000 + ..., see formulas (46)-(47) in the reference below.

			Denominators of 0/1, -1/60, 5/336, -469/21600, 6515/133056, -131672123/825552000, ...

G. C. Greubel, Table of n, a(n) for n = 1..500
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. A001620, A002206, A195189, A075266, A262235, A001067, A006953, A082633, A262382, A262383, A086279, A262384 (numerators of this series), A086280, A262386, A262387.

a := n -> denom(-Zeta(1 - 2*n)*(Psi(1, 2*n) + (Psi(0,2*n) + gamma)^2 - (Pi^2)/6)):
seq(a(n), n=1..18); # Peter Luschny, Apr 19 2018

a[n_] := Denominator[BernoulliB[2*n]*(HarmonicNumber[2*n - 1]^2 - HarmonicNumber[2*n - 1, 2])/(2*n)]; Table[a[n], {n, 1, 20}]

a(n) = denominator(bernfrac(2*n)*(sum(k=1,2*n-1,1/k)^2 - sum(k=1,2*n-1,1/k^2))/(2*n)); \\ Michel Marcus, Sep 23 2015

a(n) = denominator(B_{2n}*(H^2_{2n-1}-H^(2)_{2n-1})/(2n)), where B_n, H_n and H^(k)_n are Bernoulli, harmonic and generalized harmonic numbers respectively.

a(n) = denominator(-Zeta(1 - 2*n)*(Psi(1,2*n) + (Psi(0,2*n) + gamma)^2 - (Pi^2)/6)), where gamma is Euler's gamma and Psi is the digamma function. - Peter Luschny, Apr 19 2018

cp's OEIS Frontend

A060054 Numerators of numbers appearing in the Euler-Maclaurin summation formula.

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A090496 Ratio of numerator(Bernoulli(2*n)/(2*n)) to numerator(Bernoulli(2*n)/(2*n*(2*n-1))) for n's for which they are different.

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A141590 a(n) = numerator of Bernoulli(2*n)/(2*n + 1)!. Bisection of A120082.

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A112548 Numbers k such that the numerator of Bernoulli(k)/k is (apart from sign) prime.

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A262382 Numerators of a semi-convergent series leading to the first Stieltjes constant gamma_1.

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A262383 Denominators of a semi-convergent series leading to the first Stieltjes constant gamma_1.

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A262387 Denominators of a semi-convergent series leading to the third Stieltjes constant gamma_3.

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A090496 Ratio of numerator(Bernoulli(2n)/(2n)) to numerator(Bernoulli(2n)/(2n(2n-1))) for n's for which they are different.

A141590 a(n) = numerator of Bernoulli(2n)/(2n + 1)!. Bisection of A120082.