A238182 -id:A238182 - 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).

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).

A241215 Decimal expansion of Sum_{n>=1} H(n)^4/(n+1)^3 where H(n) is the n-th harmonic number.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 17 2014

			1.80161326804341290372948894202088843...

Philippe Flajolet and Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998), page 16.

Cf. A016627, A083680, A102886, A152648, A152649, A152651, A233033, A233090, A238166, A238167, A239168, A238169, A238181, A238182, A238183, A240264.

37/180*Pi^4*Zeta[3] - 5/6*Pi^2*Zeta[5] - 109/8*Zeta[7] // RealDigits[#, 10, 100]& // First

37/2*zeta(3)*zeta(4) - 5*zeta(2)*zeta(5) - 109/8*zeta(7) \\ Stefano Spezia, Jan 19 2025

Equals (37/2)*zeta(3)*zeta(4) - 5*zeta(2)*zeta(5) - (109/8)*zeta(7).

Equals (37/180)*Pi^4*zeta(3) - (5/6)*Pi^2*zeta(5) - (109/8)*zeta(7).

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.

A384461 Decimal expansion of Sum_{k>=1} H(k)^4/k^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

Original entry on oeis.org

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

Offset: 2

Amiram Eldar, May 30 2025

			45.83394146541655719259576578914226334887951133154848...

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2021, p. 230, eq. (4.122).

Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer International Publishing, 2019, p. 296, eq. (4.39).

Cf. A001008, A002805.
Cf. A002117, A013664.
Related constants: A152648, A152649, A152651, A218505, A233090, A238168, A238181, A238182, A241753, A244667, A253191, A256988, A345203.

RealDigits[979*Zeta[6]/24 + 3*Zeta[3]^2, 10, 120][[1]]

PARI
```
979*zeta(6)/24 + 3*zeta(3)^2
```

Equals 979*zeta(6)/24 + 3*zeta(3)^2.

A384462 Decimal expansion of Sum_{k>=1} H(k)^3/k^3, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 30 2025

			2.30095455170052503980642276989225600046997564640623...

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2021, p. 231, eq. (4.126).

Cornel Ioan Vălean, A master theorem of series and an evaluation of a cubic harmonic series, Journal of Classical Analysis, Vol. 10, No. 2 (2017), pp. 97-107.
Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer International Publishing, 2019, p. 294, eq. (4.35).

Cf. A001008, A002805.
Cf. A002117, A013664.
Related constants: A152648, A152649, A152651, A218505, A233090, A238168, A238181, A238182, A241753, A244667, A253191, A256988, A345203.

RealDigits[93*Zeta[6]/16 - 5*Zeta[3]^2/2, 10, 120][[1]]

PARI
```
93*zeta(6)/16 - 5*zeta(3)^2/2
```

Equals 93*zeta(6)/16 - 5*zeta(3)^2/2.

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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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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A241215 Decimal expansion of Sum_{n>=1} H(n)^4/(n+1)^3 where H(n) is the n-th harmonic number.

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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*(3*k+1)*H(k)^3/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

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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.

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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.