A013665 -id:A013665 - cp's OEIS frontend

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

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).

A016843 a(n) = (4n+3)^7.

Original entry on oeis.org

2187, 823543, 19487171, 170859375, 893871739, 3404825447, 10460353203, 27512614111, 64339296875, 137231006679, 271818611107, 506623120463, 897410677851, 1522435234375, 2488651484819, 3938980639167, 6060711605323, 9095120158391, 13348388671875, 19203908986159

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A004767, A013665.

A016843:=n->(4*n+3)^7; seq(A016843(n), n=0..50); # Wesley Ivan Hurt, Dec 26 2013

Table[(4n+3)^7, {n, 0, 50}] (* Wesley Ivan Hurt, Dec 26 2013 *)

a(n) = A004767(n)^7. - Wesley Ivan Hurt, Dec 26 2013

Sum_{n>=0} 1/a(n) = 127*zeta(7)/256 - 61*Pi^7/368640. - Amiram Eldar, Apr 24 2023

A017675 Numerator of sum of -6th powers of divisors of n.

Original entry on oeis.org

1, 65, 730, 4161, 15626, 23725, 117650, 266305, 532171, 101569, 1771562, 506255, 4826810, 3823625, 2281396, 17043521, 24137570, 34591115, 47045882, 32509893, 85884500, 57575765, 148035890, 97201325, 244156251, 12067025, 387952660, 244770825, 594823322

Offset: 1

N. J. A. Sloane

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

			1, 65/64, 730/729, 4161/4096, 15626/15625, 23725/23328, 117650/117649, 266305/262144, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A017676 (denominator), A013664, A013665.

[Numerator(DivisorSigma(6,n)/n^6): n in [1..20]]; // G. C. Greubel, Nov 07 2018

A017675[n_Integer] := Numerator[DivisorSigma[-6, n]]; Table[A017675[n], {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)
Table[Numerator[DivisorSigma[6, n]/n^6], {n, 1, 20}] (* G. C. Greubel, Nov 07 2018 *)

vector(20, n, numerator(sigma(n, 6)/n^6)) \\ G. C. Greubel, Nov 07 2018

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^6*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018
From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017676(n) = zeta(6) (A013664).
Dirichlet g.f. of a(n)/A017676(n): zeta(s)*zeta(s+6).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017676(k) = zeta(7) (A013665). (End)

A017677 Numerator of sum of -7th powers of divisors of n.

Original entry on oeis.org

1, 129, 2188, 16513, 78126, 23521, 823544, 2113665, 4785157, 5039127, 19487172, 9032611, 62748518, 13279647, 56979896, 270549121, 410338674, 205761751, 893871740, 645047319, 1801914272, 628461297, 3404825448, 385391585, 6103593751, 4047279411, 10465138360, 34691791

Offset: 1

N. J. A. Sloane

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

			1, 129/128, 2188/2187, 16513/16384, 78126/78125, 23521/23328, 823544/823543, 2113665/2097152, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A017678 (denominator), A013665, A013666.

[Numerator(DivisorSigma(7,n)/n^7): n in [1..20]]; // G. C. Greubel, Nov 07 2018

Table[Numerator[Total[Divisors[n]^-7]],{n,30}] (* Harvey P. Dale, Nov 29 2014 *)
Table[Numerator[DivisorSigma[7, n]/n^7], {n, 1, 20}] (* G. C. Greubel, Nov 07 2018 *)

vector(20, n, numerator(sigma(n, 7)/n^7)) \\ G. C. Greubel, Nov 07 2018

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^7*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018
From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017678(n) = zeta(7) (A013665).
Dirichlet g.f. of a(n)/A017678(n): zeta(s)*zeta(s+7).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017678(k) = zeta(8) (A013666). (End)

A321817 a(n) = Sum_{d|n, n/d odd} d^6 for n > 0.

Original entry on oeis.org

1, 64, 730, 4096, 15626, 46720, 117650, 262144, 532171, 1000064, 1771562, 2990080, 4826810, 7529600, 11406980, 16777216, 24137570, 34058944, 47045882, 64004096, 85884500, 113379968, 148035890, 191365120, 244156251, 308915840, 387952660

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A321543 - A321565, A321807 - A321836 for related sequences.

Cf. A013665.

a[n_] := DivisorSum[n, #^6 &, OddQ[n/#] &]; Array[a, 30] (* Amiram Eldar, Nov 02 2022 *)

apply( A321817(n)=sumdiv(n,d,if(bittest(n\d,0),d^6)), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^6*x^k/(1 - x^(2*k)). - Ilya Gutkovskiy, Dec 22 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 2^(6*e) and a(p^e) = (p^(6*e+6)-1)/(p^6-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^7, where c = 127*zeta(7)/896 = 0.142924... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-6)*(1-1/2^s). - Amiram Eldar, Jan 08 2023

A334668 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^7.

Original entry on oeis.org

1, 127, 277877, 35566069, 2778634972433, 2778624972433, 2288319945032390119, 292904812254103669607, 640582959329009845430509, 640582894792751053318381, 12483149055863937257566687990151, 12483148704882733411491577990151, 783299081298153288298872171970695856067

Offset: 1

Petros Hadjicostas, May 07 2020

Lim_{n -> infinity} a(n)/A334669(n) = A275710 = (63/64)*A013665.

			The first few fractions are 1, 127/128, 277877/279936, 35566069/35831808, 2778634972433/2799360000000, 2778624972433/2799360000000, ... = A334668/A334669.

Cf. A013665, A275710, A334669 (denominators).

Numerator @ Accumulate[Table[(-1)^(k + 1)/k^7, {k, 1, 13}]] (* Amiram Eldar, May 08 2020 *)

a(n) = numerator(sum(k=1, n, (-1)^(k+1)/k^7)); \\ Michel Marcus, May 08 2020

A334669 Denominator of Sum_{k=1..n} (-1)^(k+1)/k^7.

Original entry on oeis.org

1, 128, 279936, 35831808, 2799360000000, 2799360000000, 2305393332480000000, 295090346557440000000, 645362587921121280000000, 645362587921121280000000, 12576291107821424895098880000000, 12576291107821424895098880000000, 789143616376081512994335288360960000000

Offset: 1

Petros Hadjicostas, May 07 2020

Lim_{n -> infinity} A334668(n)/a(n) = A275710 = (63/64)*A013665.

			The first few fractions are 1, 127/128, 277877/279936, 35566069/35831808, 2778634972433/2799360000000, 2778624972433/2799360000000, ... = A334668/A334669.

Michael De Vlieger, Table of n, a(n) for n = 1..336

Cf. A013665, A275710, A334668 (numerators).

Denominator @ Accumulate[Table[(-1)^(k + 1)/k^7, {k, 1, 13}]] (* Amiram Eldar, May 08 2020 *)

a(n) = denominator(sum(k=1, n, (-1)^(k+1)/k^7)); \\ Michel Marcus, May 08 2020

A379814 a(n) = sigma_2(n) * sigma_3(n).

Original entry on oeis.org

1, 45, 280, 1533, 3276, 12600, 17200, 49725, 68887, 147420, 162504, 429240, 373660, 774000, 917280, 1596221, 1425060, 3099915, 2483320, 5022108, 4816000, 7312680, 6449040, 13923000, 10253901, 16814700, 16760800, 26367600, 20536380, 41277600, 28659904, 51117885

Offset: 1

Amiram Eldar, Jan 03 2025

See A379812 for more links and Ramanujan's general formula.

Srinivasa Ramanujan, Collected papers, edited by G. H. Hardy et al., Chelsea, 1962, chapter 17, pp. 133-135.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Srinivasa Ramanujan, Some formulae in the analytic theory of numbers, Messenger of Mathematics, Vol. 45 (1916), pp. 81-84.

Cf. A001157, A001158, A091260, A092344, A092344, A298754, A379812, A379813.

Cf. A002117, A013662, A013664, A013665.

a[n_] := Times @@ DivisorSigma[{2, 3}, n]; Array[a, 50]

a(n) = {my(f = factor(n)); sigma(f, 2) * sigma(f, 3);}

a(n) = A001157(n) * A001158(n).
Multiplicative with a(p^e) = (p^(2*e+2)-1) * (p^(3*e+3)-1) / ((p^2-1) * (p^3-1)).
Dirichlet g.f.: zeta(s) * zeta(s-2) * zeta(s-3) * zeta(s-5) / zeta(2*s-5).
Sum_{k=1..n}  a(k) ~ c * n^6 / 6, where c = zeta(3) * zeta(4) * zeta(6) / zeta(7) = Pi^10 * zeta(3) / (85050 * zeta(7)) = 1.31261826893951336264... .

A013683 Continued fraction for zeta(7).

Original entry on oeis.org

1, 119, 1, 3, 2, 1, 2, 1, 39, 2, 3, 12, 3, 1, 1, 1, 2, 6, 5, 1, 5, 1, 2, 1, 23, 2, 1, 5, 34, 2, 1, 1, 3, 47, 2, 1, 8, 16, 1, 4, 1, 2, 1, 1, 1, 10, 72, 1, 1, 1, 1, 1, 2, 3, 13, 1, 2, 1, 5, 1, 27, 2, 9283, 1, 36, 1, 1, 1, 1, 3, 3, 23, 27, 5, 2, 4, 1, 3, 16, 1, 4

Offset: 0

N. J. A. Sloane

Cf. A013665 (decimal expansion).

Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.

ContinuedFraction[Zeta[7],100] (* Harvey P. Dale, Sep 13 2020 *)

Offset changed by Andrew Howroyd, Jul 09 2024

A309927 Decimal expansion of Pi^7/Zeta(7).

Original entry on oeis.org

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

Offset: 4

Seiichi Manyama, Aug 23 2019

Index entries for zeta function.

Cf. A013665, A092735, A308637.

RealDigits[Pi^7/Zeta[7], 10, 100][[1]] (* Amiram Eldar, Aug 24 2019 *)

PARI
```
Pi^7/zeta(7)
```

Pi^7/Zeta(7) = A092735/A013665.

More terms from Amiram Eldar, Aug 24 2019

cp's OEIS Frontend

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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A016843 a(n) = (4n+3)^7.

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A017675 Numerator of sum of -6th powers of divisors of n.

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Magma

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A017677 Numerator of sum of -7th powers of divisors of n.

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A321817 a(n) = Sum_{d|n, n/d odd} d^6 for n > 0.

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A334668 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^7.

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A334669 Denominator of Sum_{k=1..n} (-1)^(k+1)/k^7.

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