A238168 -id:A238168 - cp's OEIS frontend

A238181 Decimal expansion of sum_(n>=1) H(n)^2/n^3 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,3)).

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

Offset: 1

Jean-François Alcover, Feb 19 2014

			1.6519427927044986239626937611144940161...

Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 24.

Cf. A152648, A152649, A152651, A218505, A238168, A238182, A238183.

7/2*Zeta[5] - Zeta[2]*Zeta[3] // RealDigits[#, 10, 100]& // First

7/2*zeta(5) - zeta(2)*zeta(3) \\ Stefano Spezia, May 22 2025

7/2*zeta(5) - zeta(2)*zeta(3).

A238182 Decimal expansion of Sum_{n>=1} H(n)^2/n^4 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,4)).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 19 2014

No closed form of S(2,2q) is known to date, except for S(2,2) (A218505) and S(2,4) (this sequence).

			1.221879945319880138518806475290994812569...

Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 24.

Cf. A001008/A002805, A152648, A152649, A152651, A218505, A238168, A238181, A238183.

97/24*Zeta[6] - 2*Zeta[3]^2 // RealDigits[#, 10, 100]& // First

97/24*zeta(6) - 2*zeta(3)^2.

A238166 Decimal expansion of sum_(n>=1) H(n,2)/n^4 where H(n,2) = A007406(n)/A007407(n) is the n-th harmonic number of order 2.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 19 2014

			1.1058264444388178540088457688766809845497962424196...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 16.

Cf. A007406, A007407, A152648, A152649, A152651, A238167, A238168, A238169.

RealDigits[Zeta[3]^2 - 1/3*Zeta[6], 10, 100][[1]]

zeta(3)^2-Pi^6/2835 /* Michel Marcus, Jul 04 2014 */

Equals zeta(3)^2 - zeta(6)/3.

A238167 Decimal expansion of sum_(n>=1) H(n,3)/n^5 where H(n,3) = A007408(n)/A007409(n) is the n-th harmonic number of order 3.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 19 2014

			1.046924401724676082345723014222792329619598402264...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 16.

Cf. A007408, A007409, A152648, A152649, A152651, A238166, A238168, A238169.

RealDigits[5*Zeta[2]*Zeta[5] +2*Zeta[3]*Zeta[4] -10*Zeta[7],10,100][[1]]

5*zeta(2)*zeta(5) + 2*zeta(3)*zeta(4) - 10*zeta(7) \\ G. C. Greubel, Dec 30 2017

Equals 5*zeta(2)*zeta(5) + 2*zeta(3)*zeta(4) - 10*zeta(7).

A238169 Decimal expansion of sum_(n>=1) H(n)^3/n^4 where H(n) is the n-th harmonic number.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 19 2014

			1.38146831050385237300478512040662269993...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 16.

Cf. A152648, A152649, A152651, A238166, A238167, A238168.

RealDigits[(231/16)*Zeta[7] - (51/4)*Zeta[3]*Zeta[4] + 2*Zeta[2]*Zeta[5], 10, 100][[1]] (* G. C. Greubel, Dec 30 2017 *)

(231/16)*zeta(7) - (51/4)*zeta(3)*zeta(4) + 2*zeta(2)*zeta(5) \\ G. C. Greubel, Dec 30 2017

Equals (231/16)*Zeta(7) - (51/4)*Zeta(3)*Zeta(4) + 2*Zeta(2)*Zeta(5).

A238183 Decimal expansion of sum_(n>=1) H(n)^2/n^7 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,7)).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 19 2014

			1.019483497494382286206496675928126515...

Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 24.

Cf. A152648, A152649, A152651, A218505, A238168, A238181, A238182.

Zeta[3]^3/3 - 5/2*Zeta[4]*Zeta[5] - 7/2*Zeta[3]*Zeta[6] - Zeta[2]*Zeta[7] + 55/6*Zeta[9] // RealDigits[#, 10, 100]& // First

zeta(3)^3/3-5/2*zeta(4)*zeta(5)-7/2*zeta(3)*zeta(6)-zeta(2)*zeta(7)+55/6*zeta(9).

A384457 Decimal expansion of Sum_{k>=1} H(k)^3/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 30 2025

			3.59342794177494296025518240703339219591695480351933...

K. Ramachandra and R. Sitaramachandrarao, On series, integrals and continued fractions - II, Madras Univ. J., Sect. B, 51 (1988), pp. 181-198.

K. Ramachandra, On series integrals and continued fractions I, Hardy-Ramanujan Journal, Vol. 4 (1981), pp. 1-11.
K. Ramachandra, On series, integrals and continued fractions, III, Acta Arithmetica, Vol. 99, No. 3 (2001), pp. 257-266.

Cf. A001008, A002805.
Cf. A000796, A002117, A002162, A352769.
Related constants: A152648, A152649, A152651, A218505, A233090, A238168, A238181, A238182, A241753, A244667, A253191, A256988, A345203.

RealDigits[Zeta[3] + (Pi^2*Log[2] + Log[2]^3)/3, 10, 120][[1]]

PARI
```
zeta(3) + (Pi^2*log(2) + log(2)^3)/3
```

Equals zeta(3) + (Pi^2*log(2) + log(2)^3)/3.

A384458 Decimal expansion of Sum_{k>=1} (-1)^(k+1)*H(k)^3/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, May 30 2025

			0.27412574654925297067883303678750470762654489295575...

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2021, p. 245, eq. (4.149).
K. Ramachandra and R. Sitaramachandrarao, On series, integrals and continued fractions - II, Madras Univ. J., Sect. B, 51 (1988), pp. 181-198.

K. Ramachandra, On series integrals and continued fractions I, Hardy-Ramanujan Journal, Vol. 4 (1981), pp. 1-11.
K. Ramachandra, On series, integrals and continued fractions, III, Acta Arithmetica, Vol. 99, No. 3 (2001), pp. 257-266.

Cf. A001008, A002805.
Cf. A000796, A002117, A002162, A013662.
Related constants: A152648, A152649, A152651, A218505, A233090, A238168, A238181, A238182, A241753, A244667, A253191, A256988, A345203.

RealDigits[(Pi*Log[2])^2/8 + 5*Zeta[4]/8 - 9*Zeta[3]*Log[2]/8 - Log[2]^4/4, 10, 120][[1]]

(Pi*log(2))^2/8 + 5*zeta(4)/8 - 9*zeta(3)*log(2)/8 - log(2)^4/4

Equals (Pi*log(2))^2/8 + 5*zeta(4)/8 - 9*zeta(3)*log(2)/8 - log(2)^4/4.

A384459 Decimal expansion of Sum_{k>=1} (-1)^k(3k+1)*H(k)^3/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, May 30 2025

			0.16440195389316542965263621650302311406441305151904...

K. Ramachandra and R. Sitaramachandrarao, On series, integrals and continued fractions - II, Madras Univ. J., Sect. B, 51 (1988), pp. 181-198.

K. Ramachandra, On series integrals and continued fractions I, Hardy-Ramanujan Journal, Vol. 4 (1981), pp. 1-11.
K. Ramachandra, On series, integrals and continued fractions, III, Acta Arithmetica, Vol. 99, No. 3 (2001), pp. 257-266.
Michael Ian Shamos, Shamos's Catalog of the Real Numbers, 2011, p. 225.

Cf. A001008, A002805, A016578.

Related constants: A152648, A152649, A152651, A218505, A233090, A238168, A238181, A238182, A241753, A244667, A253191, A256988, A345203.

Mathematica
```
RealDigits[Log[3/2]^2, 10, 120][[1]]
```
PARI
```
log(3/2)^2
```

Equals A016578^2 = log(3/2)^2 (Ramachandra, 1981).

Equals Sum_{k>=1} (-1)^(k+1)*H(k)/((k+1)*2^k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number (Shamos, 2011).

A384460 Decimal expansion of Sum_{k>=1} (-1)^(k+1)*H(k)^2/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, May 30 2025

			0.44246018937791249521879821917465633518413362702258...

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals, Springer, 2013, section 3.4, p. 148.

Cf. A001008, A002805.
Cf. A000796, A002117, A002162.
Related constants: A152648, A152649, A152651, A218505, A233090, A238168, A238181, A238182, A241753, A244667, A253191, A256988, A345203.

RealDigits[(9*Zeta[3] + 4*Log[2]^3 - Pi^2*Log[2])/12, 10, 120][[1]]

(9*zeta(3) + 4*log(2)^3 - Pi^2*log(2))/12

Equals (9*zeta(3) + 4*log(2)^3 - Pi^2*log(2))/12.

cp's OEIS Frontend

A238181 Decimal expansion of sum_(n>=1) H(n)^2/n^3 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,3)).

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A238182 Decimal expansion of Sum_{n>=1} H(n)^2/n^4 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,4)).

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A238166 Decimal expansion of sum_(n>=1) H(n,2)/n^4 where H(n,2) = A007406(n)/A007407(n) is the n-th harmonic number of order 2.

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A238167 Decimal expansion of sum_(n>=1) H(n,3)/n^5 where H(n,3) = A007408(n)/A007409(n) is the n-th harmonic number of order 3.

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A238169 Decimal expansion of sum_(n>=1) H(n)^3/n^4 where H(n) is the n-th harmonic number.

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A238183 Decimal expansion of sum_(n>=1) H(n)^2/n^7 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,7)).

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A384457 Decimal expansion of Sum_{k>=1} H(k)^3/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

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A384458 Decimal expansion of Sum_{k>=1} (-1)^(k+1)*H(k)^3/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

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A384459 Decimal expansion of Sum_{k>=1} (-1)^k(3k+1)*H(k)^3/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.