A006953 -id:A006953 - cp's OEIS frontend

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

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.

A262856 Numerators of the Nielsen-Jacobsthal series leading to Euler's constant.

Original entry on oeis.org

1, 43, 20431, 2150797323119, 9020112358835722225404403, 51551916515442115079024221439308876243677598340510141

Offset: 1

Iaroslav V. Blagouchine, Oct 03 2015

gamma = 1 - 1/12 - 43/420 - 20431/240240 - 2150797323119/36100888223400 - ..., see formula (36) in the reference below.

			Numerators of 1/12, 43/420, 20431/240240, 2150797323119/36100888223400, ...

G. C. Greubel, Table of n, a(n) for n = 1..10
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. A075266, A075267, A001620, A195189, A002657, A002790, A262235, A075266, A006953, A001067, A262858 (denominators of this series).

List(List([1..6],n->n*Sum([2^n+1..2^(n+1)],k->(-1)^(k+1)/k)),NumeratorRat); # Muniru A Asiru, Oct 29 2018

[Numerator(n*(&+[(-1)^(k+1)/k: k in [2^n+1..2^(n+1)]])): n in [1..6]]; // G. C. Greubel, Oct 28 2018

a[n_] := Numerator[n*Sum[(-1)^(k + 1)/k, {k, 2^n + 1, 2^(n + 1)}]]; Table[a[n], {n, 1, 8}]

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

a(n) = n * Sum_{k = 2^n + 1 .. 2^(n + 1)} (-1)^(k + 1)/k.

A262858 Denominators of the Nielsen-Jacobsthal series leading to Euler's constant.

Original entry on oeis.org

12, 420, 240240, 36100888223400, 236453376820564453502272320, 2225626015166235263233958200740039423756478781341512000

Offset: 1

Iaroslav V. Blagouchine, Oct 03 2015

gamma = 1 - 1/12 - 43/420 - 20431/240240 - 2150797323119/36100888223400 - ..., see formula (36) in the reference below.

			Denominators of 1/12, 43/420, 20431/240240, 2150797323119/36100888223400, ...

G. C. Greubel, Table of n, a(n) for n = 1..10
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. A075266, A075267, A001620, A195189, A002657, A002790, A262235, A075266, A006953, A001067, A262856 (numerators of this series).

List(List([1..6],n->n*Sum([2^n+1..2^(n+1)],k->(-1)^(k+1)/k)),DenominatorRat); # Muniru A Asiru, Oct 29 2018

[Denominator(n*(&+[(-1)^(k+1)/k: k in [2^n+1..2^(n+1)]])): n in [1..6]];  // G. C. Greubel, Oct 28 2018

a[n_] := Denominator[n*Sum[(-1)^(k + 1)/k, {k, 2^n + 1, 2^(n + 1)}]]; Table[a[n], {n, 1, 8}]

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

a(n) = n * Sum_{k = 2^n + 1 .. 2^(n + 1)} (-1)^(k + 1)/k.

A358625 a(n) = numerator of Bernoulli(n, 1) / n for n >= 1, a(0) = 1.

Original entry on oeis.org

1, 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, 0, 154210205991661, 0, -261082718496449122051

Offset: 0

Peter Luschny, Dec 02 2022

The rational numbers r(n) = Bernoulli(n, 1) / n are called the 'divided Bernoulli numbers'. r(n) is a p-integer for all primes p if p - 1 does not divide n. This is sometimes called 'Adams's theorem' (Ireland and Rosen). The important Kummer congruences for the Bernoulli numbers (1851) are stated in terms of the r(n).

			Rationals: 1, 1/2, 1/12, 0, -1/120, 0, 1/252, 0, -1/240, 0, 1/132, ...
Note that a(68) = -4633713579924631067171126424027918014373353 but A120082(68) = -125235502160125163977598011460214000388469.

Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, Vol. 84, Graduate Texts in Mathematics, Springer-Verlag, 2nd edition, 1990. [Prop. 15.2.4, p. 238]

Peter Luschny, Table of n, a(n) for n = 0..300
Arnold Adelberg, Shaofang Hong and Wenli Ren, Bounds of Divided Universal Bernoulli Numbers and Universal Kummer Congruences, Proceedings of the American Mathematical Society, Vol. 136(1), 2008, p. 61-71.
Bernd C. Kellner, The structure of Bernoulli numbers, arXiv:math/0411498 [math.NT], 2004.

Cf. A001067, A342318, A120080, A120082, A120084, A120086, A079612, A075180, A060054.
Cf. A164555 / A027642.
For the denominators cf. A006953, A036283, A053657, A202318.

Concatenation([1, 1], List([2..45], n-> NumeratorRat(Bernoulli(n)/(n)))); # G. C. Greubel, Sep 19 2019

[1, 1] cat [Numerator(Bernoulli(n)/(n)): n in [2..45]]; // G. C. Greubel, Sep 19 2019

A358625 := n -> ifelse(n = 0, 1, numer(bernoulli(n, 1) / n)):
seq(A358625(n), n = 0.. 40);
# Alternative:
egf := 1 + x + log(1 - exp(-x)) - log(x): ser := series(egf, x, 42):
seq(numer(n! * coeff(ser, x, n)), n = 0..40);

Join[{1, 1}, Table[Numerator[BernoulliB[n] / n], {n, 2, 45}]]

a(n) = if (n<=1, 1, numerator(bernfrac(n)/n)); \\ Michel Marcus, Feb 24 2015

a(n) = numerator(n! * [x^n](1 + x + log(1 - exp(-x)) - log(x))).
a(n) = numerator(-zeta(1 - n)) for n >= 1.
a(n) = numerator(Euler(n-1, 1) / (2*(2^n - 1))) for n >= 1.
denominator(r(2*n)) = A006953(n) for n >= 1.
denominator(r(2*n)) / 2 = A036283(n) for n >= 1.
denominator(r(2*n)) / 12 = A202318(n) for n >= 1.
denominator(r(2*n)) = (1/2) * A053657(2*n+1) / A053657(2*n-1) for n >= 1.

A164877 First bisection of A164869.

Original entry on oeis.org

0, 12, 120, 252, 240, 660, 32760, 84, 8160, 14364, 6600, 3036, 65520, 156, 24360, 429660, 16320, 204, 69090840, 228, 541200, 75852, 30360, 12972, 2227680, 3300, 82680, 43092, 48720, 20532, 3407203800, 372, 32640, 4271652, 2040, 328020, 10087262640

Offset: 0

Paul Curtz, Aug 29 2009

nonn

All entries are multiples of 12.

a(n) = A006953(n) * A111008(n), n>0.

a(n) = A164869(2n).

a(n) = A002445(n) * A005843(n).

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

A342319 a(n) = denominator(((i^n * PolyLog(1 - n, -i) + (-i)^n * PolyLog(1 - n, i))) / (4^n - 2^n)) if n > 0 and a(0) = 1. Here i denotes the imaginary unit.

Original entry on oeis.org

1, 2, 12, 56, 120, 992, 252, 16256, 240, 261632, 132, 4192256, 32760, 67100672, 12, 1073709056, 8160, 17179738112, 14364, 274877382656, 6600, 4398044413952, 276, 70368735789056, 65520, 1125899873288192, 12, 18014398375264256, 3480, 288230375614840832

Offset: 0

Peter Luschny, Mar 22 2021

For comments and references see A342318.

			r(n) = 1, 1/2, 1/12, 1/56, 1/120, 5/992, 1/252, 61/16256, 1/240, 1385/261632, 1/132, ...

Cf. A342318 (numerator), A006953, A193475.

a := n -> `if`(n = 0, 1, `if`(n::even, denom(abs(bernoulli(n))/n), 4^n - 2^n)):
seq(a(n), n=0..29);

r[s_] := If[s == 0, 1, (I^s PolyLog[1 - s, -I] + (-I)^s PolyLog[1 - s, I]) / (4^s - 2^s)]; Table[r[n], {n, 0, 29}] // Denominator

a(2*n) = A006953(n).

a(2*n+1) = A193475(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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A262386 Numerators of a semi-convergent series leading to the third Stieltjes constant gamma_3.

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A262856 Numerators of the Nielsen-Jacobsthal series leading to Euler's constant.

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A262858 Denominators of the Nielsen-Jacobsthal series leading to Euler's constant.

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A358625 a(n) = numerator of Bernoulli(n, 1) / n for n >= 1, a(0) = 1.

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