A086280 -id:A086280 - cp's OEIS frontend

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

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.

A301816 Decimal expansion of the real Stieltjes gamma function at x = 1/2.

Original entry on oeis.org

2, 7, 5, 4, 3, 4, 7, 2, 4, 5, 6, 3, 9, 2, 0, 0, 7, 9, 9, 5, 5, 2, 8, 7, 8, 7, 7, 7, 9, 7, 8, 0, 6, 8, 3, 5, 7, 9, 8, 7, 0, 2, 3, 2, 3, 8, 8, 6, 3, 0, 7, 4, 8, 7, 3, 7, 3, 3, 2, 1, 1, 4, 7, 5, 1, 3, 3, 0, 6, 3, 4, 4, 1, 7, 3, 0, 6, 4, 6, 8, 8, 2, 2, 3, 5, 9, 2

Offset: 0

Peter Luschny, Apr 09 2018

Define the real Stieltjes gamma function (this is not a standard notion) as Sti(x) = -2*Pi*I(x+1)/(x+1) where I(x) = Integral_{-infinity..+infinity} log(1/2+i*z)^x/(exp(-Pi*z) + exp(Pi*z))^2 dz and i is the imaginary unit. We look here at the real part of Sti(x).

			0.2754347245639200799552878777978068357987023238863074873733211475133063441...

Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, Journal of Number Theory, vol. 148, pp. 537-592 and vol. 151, pp. 276-277, 2015. arXiv version, arXiv:1401.3724 [math.NT], 2014.
Peter Luschny, Illustration of the real Stieltjes gamma function.

Sti(0) = A001620 (Euler's constant gamma) (cf. A262235/A075266),
Sti(1/2) = A301816,
Sti(1) = A082633 (Stieltjes constant gamma_1) (cf. A262382/A262383),
Sti(3/2) = A301817,
Sti(2) = A086279 (Stieltjes constant gamma_2) (cf. A262384/A262385),
Sti(3) = A086280 (Stieltjes constant gamma_3) (cf. A262386/A262387),
Sti(4) = A086281, Sti(5) = A086282, Sti(6) = A183141, Sti(7) = A183167,
Sti(8) = A183206, Sti(9) = A184853, Sti(10) = A184854.

Sti := x -> (-4*Pi/(x + 1))*int(log(1/2 + I*z)^(x + 1)/(exp(-Pi*z) + exp(Pi*z))^2, z=0..64): Sti(1/2): Re(evalf(%, 100)); # Note that this is an approximation which needs a larger domain of integration and higher precision if used for more values than are in the Data section.

c = -Re((4/3)*Pi*Integral_{-oo..oo} log(1/2+i*z)^(3/2)/(exp(-Pi*z)+exp(Pi*z))^2 dz).

A061203 (tau<=)_5(n).

Original entry on oeis.org

1, 6, 11, 26, 31, 56, 61, 96, 111, 136, 141, 216, 221, 246, 271, 341, 346, 421, 426, 501, 526, 551, 556, 731, 746, 771, 806, 881, 886, 1011, 1016, 1142, 1167, 1192, 1217, 1442, 1447, 1472, 1497, 1672, 1677, 1802, 1807, 1882, 1957, 1982, 1987, 2337, 2352

Offset: 1

Vladeta Jovovic, Apr 21 2001

(tau<=)_k(n) = |{(x_1,x_2,...,x_k): x_1*x_2*...*x_k <= n}|, i.e., (tau<=)_k(n) is number of solutions to x_1*x_2*...*x_k <= n, x_i > 0.
Partial sums of A061200.
Equals row sums of triangle A140705. - Gary W. Adamson, May 24 2008

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - The asymptotic ratio (100000 terms)

Cf. tau_2(n): A000005, tau_3(n): A007425, tau_4(n): A007426, tau_5(n): A061200, tau_6(n): A034695, (unordered) 2-factorizations of n: A038548, (unordered) 3-factorizations of n: A034836, A001055, (tau<=)_2(n): A006218, (tau<=)_3(n): A061201, (tau<=)_4(n): A061202, (tau<=)_6(n): A061204.

Cf. A140705.

b:= proc(k, n) option remember; uses numtheory;
     `if`(k=1, 1, add(b(k-1, d), d=divisors(n)))
    end:
a:= proc(n) option remember; `if`(n=0, 0, b(5, n)+a(n-1)) end:
seq(a(n), n=1..49);  # Alois P. Heinz, Feb 13 2022

nmax = 50;
tau4 = Table[DivisorSum[n, DivisorSigma[0, n/#]*DivisorSigma[0, #] &], {n, 1, nmax}];
Accumulate[Table[Sum[tau4[[d]], {d, Divisors[n]}], {n, nmax}]] (* Vaclav Kotesovec, Sep 10 2018 *)

(tau<=)k(n) = Sum{i=1..n} tau_k(i).
a(n) = Sum_{k=1..n} tau_{4}(k) * floor(n/k), where tau_{4} is A007426. - Enrique Pérez Herrero, Jan 23 2013
a(n) ~ n*(log(n)^4/24 + (5*g/6 - 1/6)*log(n)^3 + 10*g1^2 + (5*g^2 - 5*g/2 - 5*g1/2 + 1/2)*log(n)^2 + (10*g^3 - 10*g^2 + (5 - 20*g1)*g + 5*g1 + 5*g2/2 - 1)*log(n) + 5*g^4 - 10*g^3 + (10 - 30*g1)*g^2 + (20*g1 + 10*g2 - 5)*g - 5*g1 - 5*g2/2 - 5*g3/6 + 1), where g is the Euler-Mascheroni constant A001620 and g1, g2, g3 are the Stieltjes constants, see A082633, A086279 and A086280. - Vaclav Kotesovec, Sep 10 2018

A061204 (tau<=)_6(n).

Original entry on oeis.org

1, 7, 13, 34, 40, 76, 82, 138, 159, 195, 201, 327, 333, 369, 405, 531, 537, 663, 669, 795, 831, 867, 873, 1209, 1230, 1266, 1322, 1448, 1454, 1670, 1676, 1928, 1964, 2000, 2036, 2477, 2483, 2519, 2555, 2891, 2897, 3113, 3119, 3245, 3371, 3407, 3413

Offset: 1

Vladeta Jovovic, Apr 21 2001

(tau<=)_k(n) = |{(x_1,x_2,...,x_k): x_1*x_2*...*x_k<=n}|, i.e. (tau<=)_k(n) is number of solutions to x_1*x_2*...*x_k<=n, x_i>0.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - The asymptotic ratio (100000 terms)

Cf. tau_2(n): A000005, tau_3(n): A007425, tau_4(n): A007426, tau_5(n): A061200, tau_6(n): A034695, (unordered) 2-factorizations of n: A038548, (unordered) 3-factorizations of n: A034836, A001055, (tau<=)_2(n): A006218, (tau<=)_3(n): A061201, (tau<=)_4(n): A061202, (tau<=)_5(n): A061203.

nmax = 50; tau4 = Table[DivisorSum[n, DivisorSigma[0, n/#]*DivisorSigma[0, #] &], {n, 1, nmax}]; tau5 = Table[Sum[tau4[[d]], {d, Divisors[n]}], {n, nmax}]; Accumulate[Table[Sum[tau5[[d]], {d, Divisors[n]}], {n, nmax}]] (* Vaclav Kotesovec, Sep 10 2018 *)

(tau<=)k(n)=Sum{i=1..n} tau_k(i). a(n)=partial sums of A034695.
a(n) = Sum_{k=1..n} tau_{5}(k) * floor(n/k), where tau_{5} is A061200. - Enrique Pérez Herrero, Jan 23 2013
a(n) ~ n*(log(n)^5/120 + (g/4 - 1/24)*log(n)^4 + (5*g^2/2 - g - g1 + 1/6)*log(n)^3 + (10*g^3 - 15*g^2/2 + (3 - 15*g1)*g + 3*g1 + 3*g2/2 - 1/2)*log(n)^2 + (15*g^4 - 20*g^3 + (15 - 60*g1)*g^2 + (30*g1 + 15*g2 - 6)*g + 15*g1^2 - 6*g1 - 3*g2 - g3 + 1)*log(n) + 6*g^5 - 15*g^4 + (20 - 60*g1)*g^3 + (60*g1 + 30*g2 - 15)*g^2 + (60*g1^2 - 30*g1 - 15*g2 - 5*g3 + 6)*g - 15*g1^2 + g1*(6 - 15*g2) + 3*g2 + g3 + g4/4 - 1), where g is the Euler-Mascheroni constant A001620 and g1, g2, g3, g4 are the Stieltjes constants, see A082633, A086279, A086280 and A086281. - Vaclav Kotesovec, Sep 10 2018

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

A262386 Numerators of a semi-convergent series leading to the third Stieltjes constant gamma_3.

Original entry on oeis.org

0, 1, -17, 967, -4523, 33735311, -9301169, 127021899032857, -3546529522734769, 5633317707758173, -1935081812850766373, 779950247074296817622891, -1261508681536108282229, 350992098387568751020053498509, -17302487974885784968377519342317, 26213945071317075538702463006927083

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.

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

G. C. Greubel, Table of n, a(n) for n = 1..225
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 A262387.

Cf. A001067, A001620, A002206, A006953, A075266, A082633, A086279, A086280, A195189, A262235, A262382, A262383, A262384, A262385.

a[n_] := Numerator[-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) = numerator(-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) = numerator(-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.

A242612 Decimal expansion of the sum of the alternating series tau(4), with tau(n) = Sum_{k>0} (-1)^k*log(k)^n/k.

Original entry on oeis.org

0, 1, 7, 9, 9, 6, 9, 3, 8, 1, 0, 6, 8, 9, 1, 4, 0, 7, 7, 9, 5, 3, 6, 7, 8, 2, 1, 4, 3, 6, 1, 5, 2, 6, 2, 3, 8, 9, 8, 1, 1, 2, 3, 4, 5, 1, 3, 9, 0, 2, 3, 3, 4, 9, 2, 9, 4, 5, 0, 2, 4, 7, 9, 9, 9, 1, 3, 2, 2, 5, 6, 2, 4, 6, 3, 8, 0, 8, 5, 8, 4, 3, 0, 9, 4, 2, 9, 7, 0, 5, 9, 1, 9, 5, 1, 4, 2, 4, 2, 9, 9

Offset: 0

Jean-François Alcover, May 19 2014

			-0.017996938106891407795367821436152623898...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, chapter 2.21, p. 168.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A001620, A082633, A086279, A086280, A242494, A242611, A242613.

tau[n_] := -Log[2]^(n+1)/(n+1) + Sum[Binomial[n, k]*Log[2]^(n-k)*StieltjesGamma[k], {k, 0, n-1}]; Join[{0}, RealDigits[tau[4], 10, 100] // First]

sumalt(k=1,(-1)^k*log(k)^4/k) \\ Charles R Greathouse IV, Mar 10 2016

tau(n) = -log(2)^(n+1)/(n+1) + Sum_(k=0..n-1) (binomial(n, k)*log(2)^(n-k)*gamma(k)).

tau(4) = gamma*log(2)^4 - (1/5)*log(2)^5 + 4*log(2)^3*gamma(1) + 6*log(2)^2*gamma(2) + 4*log(2)*gamma(3).

A242613 Decimal expansion of the sum of the alternating series tau(5), with tau(n) = Sum_{k>0} (-1)^k*log(k)^n/k.

Original entry on oeis.org

0, 2, 4, 5, 1, 4, 9, 0, 7, 6, 5, 6, 4, 0, 9, 7, 8, 2, 9, 0, 7, 4, 2, 2, 8, 0, 0, 6, 8, 6, 1, 3, 7, 1, 1, 0, 2, 8, 7, 5, 7, 0, 7, 0, 9, 2, 3, 7, 9, 1, 5, 0, 3, 7, 4, 2, 9, 0, 5, 1, 1, 2, 7, 2, 9, 8, 3, 7, 8, 8, 0, 0, 9, 9, 7, 5, 5, 3, 3, 5, 8, 9, 1, 5, 4, 6, 6, 2, 9, 4, 6, 0, 6, 2, 9, 3, 7, 4, 1, 7, 8

Offset: 0

Jean-François Alcover, May 19 2014

			-0.02451490765640978290742280068613711...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, chapter 2.21, p. 168.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A001620, A082633, A086279, A086280, A086281, A242494, A242611, A242612.

tau[n_] := -Log[2]^(n+1)/(n+1) + Sum[Binomial[n, k]*Log[2]^(n-k)*StieltjesGamma[k], {k, 0, n-1}]; Join[{0}, RealDigits[tau[5], 10, 100] // First]

tau(n) = -log(2)^(n+1)/(n+1) + Sum_(k=0..n-1) (binomial(n, k)*log(2)^(n-k)*gamma(k)).

tau(5) = gamma*log(2)^5 - (1/6)*log(2)^6 + 5*log(2)^4*gamma(1) + 10*log(2)^3*gamma(2) + 10*log(2)^2*gamma(3) + 5*log(2)*gamma(4).

cp's OEIS Frontend

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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A301816 Decimal expansion of the real Stieltjes gamma function at x = 1/2.

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A061203 (tau<=)_5(n).

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A061204 (tau<=)_6(n).

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

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

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