A152648 -id:A152648 - cp's OEIS frontend

A060805 Numerators of special continued fraction for 2*zeta(3).

2, 1, 2, 1, 4, 2, 6, 4, 9, 6, 12, 9, 16, 12, 20, 16, 25, 20, 30, 25, 36, 30, 42, 36, 49, 42, 56, 49, 64, 56, 72, 64, 81, 72, 90, 81, 100, 90, 110, 100, 121, 110, 132, 121, 144, 132, 156, 144, 169, 156, 182, 169, 196, 182, 210, 196, 225, 210, 240, 225, 256, 240, 272, 256

Offset: 1

N. J. A. Sloane, Apr 29 2001

Y. V. Nesterenko, A few remarks on zeta(3), Mathematical Notes, 59 (No. 6, 1996), 625-636.

Y. V. Nesterenko, Zeta(3) and recurrence relations.
Yu. V. Nesterenko, A few remarks on zeta(3), Math. Notes 59 (1996) 625-636. [From _R. J. Mathar_, Jul 31 2010]
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,-1,1).

Cf. A060804, A060806, A060807, A060808, A008733.

Cf. A152648 (2*zeta(3)).

A060805 := proc(n) local nshf,k ; if n <= 2 then op(n,[2,1]) ; else nshf := n-1 ; k := floor(nshf/4) ; if nshf mod 4 = 1 then k*(k+1) ; elif nshf mod 4 = 0 then (k+1)^2 ; elif nshf mod 4 = 2 then (k+1)*(k+2) ; else (k+1)^2 ; end if; end if; end proc: seq(A060805(n),n=1..80) ; # R. J. Mathar, Jul 31 2010

Join[{2, 1}, LinearRecurrence[{1, 1, -1, 1, -1, -1, 1}, {2, 1, 4, 2, 6, 4, 9}, 100]] (* Jean-François Alcover, Apr 01 2020 *)

a(n) = A008733(n-1), n>2. - R. J. Mathar, Jul 31 2010

More terms from R. J. Mathar, Jul 31 2010

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).

A233033 Decimal expansion of sum_(n=1..infinity) (-1)^(n-1)*H(n)/n^3 where H(n) is the n-th harmonic number.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Dec 03 2013

			0.859247157928590615539909939475759980712884350860414926760520689766...

Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 32.

Cf. A076788 (same alternating sum with denominator n), A152648 (non-alternating sum with denominator n^2), A152649 (non-alternating sum with denominator n^3).

RealDigits[ 11*Pi^4/360 + 1/12*Pi^2*Log[2]^2 - Log[2]^4/12 - 2*PolyLog[4, 1/2] - 7/4*Log[2]*Zeta[3], 10, 100] // First

11*Pi^4/360 + Pi^2*log(2)^2/12 - log(2)^4/12 - 2*polylog(4, 1/2) - 7*log(2)*zeta(3)/4 \\ Charles R Greathouse IV, Aug 27 2014

Equals 11*Pi^4/360 +1/12*Pi^2*log(2)^2 -log(2)^4/12 -2*Li4(1/2) -7/4*log(2)*zeta(3).

Also, equals 1/2*integral_{z=0..1} (log(z)^2*log(1+z)) / (z*(1+z)) dz.

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).

A351164 Decimal expansion of gamma * BesselI(0,2) + BesselK(0,2).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Feb 03 2022

			1.4297062187372083131867465655452809577372778968399203468724...

Cf. A001008, A001044, A001620, A002805, A070910, A152648, A347952.

RealDigits[EulerGamma BesselI[0, 2] + BesselK[0, 2], 10, 110] [[1]]

Equals Sum_{k>=1} H(k) / (k!)^2, where H(k) is the k-th harmonic number.

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).

A365251 Decimal expansion of the absolute value of psi^(4)(1), the fourth derivative of the digamma function at 1.

Original entry on oeis.org

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

Offset: 2

R. J. Mathar, Aug 29 2023

			psi^(4)(1) = -24.88626612344087823195277167496882...

Cf. A013663, A231535 (3rd deriv), A152648 (2nd deriv), A013661 (1st deriv).

Maple
```
evalf(Psi(4,1)) ;
```

RealDigits[24*Zeta[5], 10, 100][[1]] (* Amiram Eldar, Aug 29 2023 *)

psi''''(1) \\ Michel Marcus, Aug 29 2023

Equals 24*A013663.

A377636 Decimal expansion of 2*zeta(3)/Pi^2 - 11/18 + log(Pi)/3.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 03 2024

			0.01405317397250177984537691644248000594732436155378...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 44.

Cf. A000302, A002117, A002388, A005408, A005843, A053510, A152648.

RealDigits[2Zeta[3]/Pi^2-11/18+Log[Pi]/3,10,100][[1]]

Equals Sum_{k>=1} zeta(2*k)/(k*(k + 1)*(2*k + 1)*(2*k + 3)*2^(2*k)) [Wilton] (see Finch).

A379561 a(n) = A003418(n+1)*H(n), where H(n) = 1 + 1/2 + ... + 1/n is the n-th harmonic number.

Original entry on oeis.org

2, 9, 22, 125, 137, 1029, 2178, 6849, 7129, 81191, 83711, 1118273, 1145993, 1171733, 2391514, 41421503, 42142223, 813635157, 825887397, 837527025, 848612385, 19761458895, 19994251455, 101086721625, 102157567401, 309561680403, 312536252003, 9146733078187

Offset: 1

Miko Labalan, Dec 26 2024

nonn

abs(log(a(n)) - n - log(log(n))) < c*sqrt(n)*log(n)^(-1/2), where constant c = (2+A206431)*Pi/4. This also gives the upper bound of the squared error, (log(a(n)) - n - log(log(n)))^2 < (c^2)*n*log(n)^(-1).
A slightly better absolute error bound could be achieved by using the imaginary part of the nontrivial zeros of the Riemann zeta function, (zetazero(n)-1/2)/sqrt(-1) ~ (2*Pi)*n*LambertW(n/exp(1))^(-1). That bound would be, abs(log(a(n)) - n - log(log(n))) < sqrt(k)*sqrt(n)*LambertW(n/exp(1))^(-1/2), where constant k = 4*Pi/(1+2*A206431). This also gives the upper bound of the squared error, (log(a(n)) - n - log(log(n)))^2 < k*n*LambertW(n/exp(1))^(-1). The midline of the squared error would run along (4/(4+A206431))*n*LambertW(n/exp(1))^(-1).
Another slightly better absolute error bound but without relying on the properties of the zeta zeros would be, abs(log(a(n)) - n - log(log(n))) < n^(3/(9-10*A077761)).
log(a(n))-c*sqrt(n)*log(n)^(-1/2) is a lower bound of sigma_1(n) = A000203(n). Such that, n+log(log(n))-c*sqrt(n)*log(n)^(-1/2) < sigma_1(n) < H(n)+exp(H(n))*log(H(n)).
a(n) gives the total number of ordered pairs (k,m) where k in set {1,2,...,n}, m in set {1,2,...,A003418(n+1)}, and k divides m. Example: For n = 3, there are 22 ordered pairs (k,m) where k is {1,2,3} and m is a multiple of k up to 12. For k = 1, every m is a multiple of 1, m is {1,2,3,4,5,6,7,8,9,10,11,12} so there are 12 pairs. For k = 2, every m is a multiple of 2, m is {2,4,6,8,10,12} so there are 6 pairs. For k = 3, every m is a multiple of 3, m is {3,6,9,12} so there are 4 pairs. So the total ordered pairs is 12 + 6 + 4 = 22 = a(3). Each ordered pair (k,m) also represents an edge in a bipartite graph. Counting all such pairs gives the total number of edges in a graph.

			a(n)/A025558(n) = [ 2/1, 9/4, 22/9, 125/48, 137/50, 1029/360, 2178/735, ... ]
To evaluate the integral:
For n = 1: Integral_{x=0..1} Li_1(x^(1/2))/x^(1/2) dx = Integral_{x=0..1} -log(1-x^(1/2))/x^(1/2) dx = -2 * -(Sum_{x=1..oo} 1/(x*(x+1))) = -2 * -1 = 2.
For n = 2: Integral_{x=0..1} Li_1(x^(1/3))/x^(1/3) dx = Integral_{x=0..1} -log(1-x^(1/3))/x^(1/3) dx = -3 * -(Sum_{x=1..oo} 1/(x*(x+2))) = -3 * -((1/2)*(1+1/2)) = -3 * -3/4 = 9/4.
For n = 3: Integral_{x=0..1} Li_1(x^(1/4))/x^(1/4) dx = Integral_{x=0..1} -log(1-x^(1/4))/x^(1/4) dx = -4 * -(Sum_{x=1..oo} 1/(x*(x+3))) = -4 * -((1/3)*(1+1/2+1/3)) = -4 * -11/18 = 22/9.

Guilherme França and André LeClair, Statistical and other properties of Riemann zeros based on an explicit equation for the n-th zero on the critical line, arXiv:1307.8395 [math.NT], 2014.
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (6) (2002) 534-543. arXiv preprint, arXiv:math/0008177 [math.NT], 2000-2001.

Cf. A001008/A002805 (harmonic numbers).
Cf. A003418 (lcm).
Cf. A025558 (denominator).
Cf. A193758 (very similar sequence).
Cf. A000203, A025529, A027457, A077761, A152648, A206431.

a(n) = lcm(vector(n+1, i, i))*sum(i=1, n, 1/i); \\ Michel Marcus, Dec 28 2024

a(n) = A025558(n)*(Integral_{x=0..1} Li_1(x^(1/(n+1)))/x^(1/(n+1)) dx).
a(n) = A025558(n) + A027457(n+1).
Integral_{x=0..1} Li_1(x^(1/(n+1)))/x^(1/(n+1)) dx = ((n+1)/n)*H(n) = a(n)/A025558(n).
((n+1)/n)*H(n) ~ log(n) + gamma + (log(n)+gamma+1/2)/n + O(1/n^2).
log(a(n)) ~ n + log(log(n)) + O(c*sqrt(n)*log(n)^(-1/2)), (See comments for constant c).
G.f. for ((n+1)/n)*H(n): G(x) = Li_2(x)+(1/2)*log(1-x)^2-log(1-x)/(1-x), the lim_{x->oo} G(x) = -zeta(2).
Hyperbolic l.g.f. for ((n+1)/n)*H(n): LH(x) = Li_2(x)+(1/2)*log(1-x)^2+Li_3(x)-Li_3(1-x)+Li_2(1-x)*log(1-x)+(1/2)*log(x)*log(1-x)^2+zeta(3), the Integral_{x=0..1} LH(x) dx = 2*zeta(3) = A152648.
Dirichlet g.f. for ((n+1)/n)*H(n): zeta(s+1)*(zeta(s)+zeta(s+2)).

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.

cp's OEIS Frontend

A060805 Numerators of special continued fraction for 2*zeta(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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A233033 Decimal expansion of sum_(n=1..infinity) (-1)^(n-1)*H(n)/n^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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A351164 Decimal expansion of gamma * BesselI(0,2) + BesselK(0,2).

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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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A365251 Decimal expansion of the absolute value of psi^(4)(1), the fourth derivative of the digamma function at 1.

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A377636 Decimal expansion of 2*zeta(3)/Pi^2 - 11/18 + log(Pi)/3.

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