A244667 -id:A244667 - cp's OEIS frontend

A256988 Decimal expansion of Sum_{k>=1} H(k)^3/k^2 where H(k) is the k-th harmonic number.

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

Offset: 2

Jean-François Alcover, Apr 14 2015

			12.346581901730995381510740306055467252652960661679262328437749...

Alois Panholzer and Helmut Prodinger, Computer-free evaluation of an infinite double sum via Euler sums Séminaire Lotharingien de Combinatoire 55 (2005), Article B55a
Eric Weisstein's MathWorld, Harmonic Number.

Cf. A002117, A013661, A013663, A152648, A152651, A238181, A244667, A256987.

RealDigits[10*Zeta[5] + (Pi^2/6)*Zeta[3], 10, 104] // First

10*zeta(5) + zeta(2)*zeta(3) \\ Michel Marcus, Apr 14 2015

Equals 10*zeta(5) + zeta(2)*zeta(3) or, 10*zeta(5) + (Pi^2/6)*zeta(3).

A244675 Decimal expansion of sum_(n>=1) (H(n)^3/(n+1)^3) where H(n) is the n-th harmonic number.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 04 2014

			0.17758688422659116882105255543380545223004150911094072394667346832845...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 27.

Cf. A001008, A002805, A002117, A013663, A013665, A244667, A244674.

SetDefaultRealField(RealField(100)); R:= RealField(); L:=RiemannZeta(); -11/120*Pi(R)^4*Evaluate(L,3) + 1/3*Pi(R)^2*Evaluate(L,5) + 119/16*Evaluate(L,7); // G. C. Greubel, Aug 31 2018

RealDigits[119/16*Zeta[7] - 33/4*Zeta[3]*Zeta[4] + 2*Zeta[2]*Zeta[5], 10, 103] // First

default(realprecision, 100);  -11/120*Pi^4*zeta(3) + 1/3*Pi^2*zeta(5) + 119/16*zeta(7) \\ G. C. Greubel, Aug 31 2018

Equals -11/120*Pi^4*zeta(3) + 1/3*Pi^2*zeta(5) + 119/16*zeta(7).

A244676 Decimal expansion of sum_(n>=1) (H(n)^3/(n+1)^6) where H(n) is the n-th harmonic number.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 04 2014

			0.02289126788224074913774364071997743746511359015190275216397993401922...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 27.

Cf. A001008, A002805, A002117, A013663, A013665, A013667, A244667, A244674, A244675.

SetDefaultRealField(RealField(100)); R:= RealField(); L:=RiemannZeta(); -(37/7560)*Pi(R)^6*Evaluate(L,3) + Evaluate(L,3)^3 - (11/120)*Pi(R)^4*Evaluate(L,5) + Pi(R)^2*Evaluate(L,7)/2 + (197/24)*Evaluate(L,9); // G. C. Greubel, Aug 31 2018

RealDigits[197/24*Zeta[9] - 33/4*Zeta[4]*Zeta[5] - 37/8*Zeta[3]*Zeta[6] + Zeta[3]^3 + 3*Zeta[2]*Zeta[7], 10, 104] // First // Prepend[#, 0]&

default(realprecision, 100);  -37/7560*Pi^6*zeta(3) + zeta(3)^3 - 11/120*Pi^4*zeta(5) + 1/2*Pi^2*zeta(7) + 197/24*zeta(9) \\ G. C. Greubel, Aug 31 2018

Equals -37/7560*Pi^6*zeta(3) + zeta(3)^3 - 11/120*Pi^4*zeta(5) + 1/2*Pi^2*zeta(7) + 197/24*zeta(9).

A247669 Decimal expansion of A_3 = Sum_{n >= 1} H(n)^2/((2n-1)(2n)(2n+1))^3, where H(n) is the n-th harmonic number.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 22 2014

			0.00464045020357812398918052923623454672526939078320273923...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) p. 34.

Cf. A244667, A244674, A244675, A244676.

SetDefaultRealField(RealField(100)); R:= RealField(); L:=RiemannZeta(); 4*Log(2)^3 + ((7*Evaluate(L,3))/8 - 35/4)*Log(2)^2 - ((-(1/8))*9*Evaluate(L,2) + (7*Evaluate(L,3))/8 + (45*Evaluate(L,4))/32 - 12)*Log(2) - Evaluate(L,2)/4 - (41*Evaluate(L,3))/8 - (3/32)*Evaluate(L,2)*Evaluate(L,3) + (45*Evaluate(L,4))/64 + (17*Evaluate(L,5))/32; // G. C. Greubel, Aug 31 2018

A3 = 4*Log[2]^3 + ((7*Zeta[3])/8 - 35/4)*Log[2]^2 - ((-(1/8))*9*Zeta[2] + (7*Zeta[3])/8 + (45*Zeta[4])/32 - 12)*Log[2] - Zeta[2]/4 - (41*Zeta[3])/8 - (3/32)*Zeta[2]*Zeta[3] + (45*Zeta[4])/64 + (17*Zeta[5])/32; Join[{0, 0}, RealDigits[A3, 10, 102] // First]

4*log(2)^3 + ((7*zeta(3))/8 - 35/4)*log(2)^2 - ((-(1/8))*9*zeta(2) + (7*zeta(3))/8 + (45*zeta(4))/32 - 12)*log(2) - zeta(2)/4 - (41*zeta(3))/8 - (3/32)*zeta(2)*zeta(3) + (45*zeta(4))/64 + (17*zeta(5))/32 \\ Michel Marcus, Sep 22 2014

A_3 = 4*log(2)^3 + ((7*zeta(3))/8 - 35/4)*log(2)^2 - ((-(1/8))*9*zeta(2) + (7*zeta(3))/8 + (45*zeta(4))/32 - 12)*log(2) - zeta(2)/4 - (41*zeta(3))/8 - (3/32)*zeta(2)*zeta(3) + (45*zeta(4))/64 + (17*zeta(5))/32.

A247670 Decimal expansion of Sum_{n >= 0} (-1)^n*H(n)/(2n+1)^3, where H(n) is the n-th harmonic number.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 22 2014

			-0.02857417063624359099908429512504431088603018691486...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) p. 34.

Cf. A244667, A244674, A244675, A244676, A247669.

s = -(Pi^3/16)*Log[2] - (7*Pi/16)*Zeta[3] + (1/512)*(PolyGamma[3, 1/4] - PolyGamma[3, 3/4]); Join[{0}, RealDigits[s, 10, 101] // First]

Equals -(Pi^3/16)*log(2) - (7*Pi/16)*zeta(3) + (1/512)*(PolyGamma(3, 1/4) - PolyGamma(3, 3/4)), where PolyGamma(n,z) gives the n-th derivative of the digamma function Psi^(n)(z).

A256987 Decimal expansion of Sum_{k>=1} H(k)*H(k,2)/k^2 where H(k) is the k-th harmonic number and H(k,2) the k-th harmonic number of order 2.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 14 2015

			3.01423210544066604452845092794215974013923238616204702067...

Alois Panholzer and Helmut Prodinger, Computer-free evaluation of an infinite double sum via Euler sums, Séminaire Lotharingien de Combinatoire 55 (2005), Article B55a
Eric Weisstein's MathWorld, Harmonic Number.

Cf. A002117, A013661, A013663, A152648, A152651, A238181, A244667, A256988.

RealDigits[Zeta[5] + (Pi^2/6)*Zeta[3], 10, 105] // First

zeta(5) + zeta(2)*zeta(3) \\ Michel Marcus, Apr 14 2015

zeta(5) + zeta(2)*zeta(3) = zeta(5) + (Pi^2/6)*zeta(3).

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

A256988 Decimal expansion of Sum_{k>=1} H(k)^3/k^2 where H(k) is the k-th harmonic number.

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A244675 Decimal expansion of sum_(n>=1) (H(n)^3/(n+1)^3) where H(n) is the n-th harmonic number.

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A244676 Decimal expansion of sum_(n>=1) (H(n)^3/(n+1)^6) where H(n) is the n-th harmonic number.

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A247669 Decimal expansion of A_3 = Sum_{n >= 1} H(n)^2/((2n-1)*(2n)*(2n+1))^3, where H(n) is the n-th harmonic number.

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A247670 Decimal expansion of Sum_{n >= 0} (-1)^n*H(n)/(2n+1)^3, where H(n) is the n-th harmonic number.

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A256987 Decimal expansion of Sum_{k>=1} H(k)*H(k,2)/k^2 where H(k) is the k-th harmonic number and H(k,2) the k-th harmonic number of order 2.

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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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A247669 Decimal expansion of A_3 = Sum_{n >= 1} H(n)^2/((2n-1)(2n)(2n+1))^3, where H(n) is the n-th harmonic number.

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.