A152648 -id:A152648 - cp's OEIS frontend

A233090 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*H(n)/n^2, where H(n) is the n-th harmonic number.

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

Offset: 0

Jean-François Alcover, Dec 04 2013, after the comment by Peter Bala about A233033

			0.7512855644747464283748363509446562442281164327128118011201697220886...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.

R. Barbieri, J. A. Mignaco, and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972) 824-864, table II (13)
Philippe Flajolet and Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998), page 32.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (C5.15)
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 561.

Cf. A002117 (zeta(3)), A197070 (3*zeta(3)/4), A233091 (7*zeta(3)/8), A076788 (alternating sum with denominator n), A152648 (non-alternating sum with denominator n^2), A152649 (non-alternating sum with denominator n^3), A233033 (alternating sum with denominator n^3).

RealDigits[ 5*Zeta[3]/8, 10, 100] // First

Equals 5*zeta(3)/8.
Equals -Integral_{x=0..1} (log(1+x)*log(1-x)/x)*dx. - Amiram Eldar, May 06 2023
Equals Sum_{m>=1} Sum_{n>=1} (-1)^(m-1)/(m*n*(m + n)) (see Finch). - Stefano Spezia, Nov 02 2024

A238168 Decimal expansion of sum_(n>=1) H(n)^2/n^5 where H(n) is the n-th harmonic number.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 19 2014

			1.091882588664530085165782130699273873...

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, A238169.

RealDigits[6*Zeta[7] -Zeta[2]*Zeta[5] -(5/2)*Zeta[3]*Zeta[4],10,100][[1]]

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

Equals 6*zeta(7) - zeta(2)*zeta(5) - 5/2*zeta(3)*zeta(4).

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)).

Original entry on oeis.org

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.

A102354 a(n) is the number of ways n can be written as k^2 * j, 0 < j <= k.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0

Offset: 1

Leroy Quet, Feb 21 2005

nonn

Sum_{n>0} a(n)/n = 2*zeta(3). See A152648.

			a(18) = 1 because 18 = k^2 * j, j <= k, in one way: k=3, j=2.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A102448, A104020, A104022, A104024, A152648.

t = Sort[ Flatten[ Table[k^2*j, {k, 11}, {j, k}]]]; Table[ Count[t, n], {n, 105}] (* Robert G. Wilson v, Feb 22 2005 *)

A102354(n) = sumdiv(n,d,(issquare(d) && (sqrtint(d) >= (n/d)))); \\ Antti Karttunen, Aug 27 2017

a(n) >= A102448(n). - Antti Karttunen, Aug 27 2017

More terms from Robert G. Wilson v, Feb 22 2005

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

Original entry on oeis.org

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).

A060804 Continued fraction for 2*zeta(3).

Original entry on oeis.org

2, 2, 2, 9, 3, 10, 1, 4, 18, 1, 3, 5, 3, 1, 3, 3, 1, 1, 5, 3, 8, 1, 2, 1, 62, 1, 1, 1, 3, 2, 2, 1, 1, 5, 3, 1, 8, 2, 2, 34, 7, 1, 1, 5, 1, 2, 3, 3, 14, 9, 214, 11, 8, 23, 1, 8, 2, 10, 2, 2, 2, 1, 1, 6, 1, 8, 2, 1, 9, 2, 1, 11, 1, 3, 3, 4, 1, 28, 6, 1, 28, 1, 15, 1, 1, 1, 2

Offset: 0

N. J. A. Sloane, Apr 29 2001

			2.404113806319188570799476323... = 2 + 1/(2 + 1/(2 + 1/(9 + 1/(3 + ...)))). - _Harry J. Smith_, Jul 12 2009

Harry J. Smith, Table of n, a(n) for n = 0..19999
Y. V. Nesterenko, Zeta(3) and recurrence relations.
Y. V. Nesterenko, A few remarks on zeta(3), Mathematical Notes, 59 (No. 6, 1996), 625-636.

Cf. A060805, A060806, A060807, A060808 (convergents).

Cf. A152648 (decimal expansion).

Digits := 100: t1 := evalf(2*Zeta(3)); convert(t1,confrac);

ContinuedFraction[2 Zeta[3],90] (* Harvey P. Dale, Apr 25 2025 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(2*zeta(3)); for (n=1, 20000, write("b060804.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Jul 12 2009

Offset changed by Andrew Howroyd, Jul 10 2024

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).

cp's OEIS Frontend

A233090 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*H(n)/n^2, where H(n) is the n-th harmonic number.

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A238168 Decimal expansion of sum_(n>=1) H(n)^2/n^5 where H(n) is the n-th harmonic number.

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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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A102354 a(n) is the number of ways n can be written as k^2 * j, 0 < j <= k.

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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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A060804 Continued fraction for 2*zeta(3).

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