A241753 -id:A241753 - cp's OEIS frontend

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.

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

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.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.