A013665 -id:A013665 - cp's OEIS frontend

A160897 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 8.

1, 127, 1093, 8128, 19531, 138811, 137257, 520192, 796797, 2480437, 1948717, 8883904, 5229043, 17431639, 21347383, 33292288, 25646167, 101193219, 49659541, 158747968, 150021901, 247487059, 154764793, 568569856, 305171875, 664088461, 580865013, 1115624896

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^7 such that the quotient group Z^7 / L is C_n. - Álvar Ibeas, Oct 30 2015

Enrique Pérez Herrero and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..5000 from Enrique Pérez Herrero)
Jin Ho Kwak and Jaeun Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
Index to Jordan function ratios J_k/J_1.

Column 7 of A263950.

Cf. A000010, A000203, A008683, A013664, A013665, A069092.

A160897 := proc(n)
    add(numtheory[mobius](n/d)*d^7,d=numtheory[divisors](n)) ;
    %/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
    printf("%d %d\n",n,A160897(n)) ;
end do: # R. J. Mathar, Mar 14 2016

A160897[n_]:=DivisorSum[n, MoebiusMu[n/# ]*#^(8 - 1)/EulerPhi[n] &] (* Enrique Pérez Herrero, Oct 27 2010 *)
f[p_, e_] := p^(6*e - 6) * (p^7-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Nov 08 2022 *)

vector(30, n, sumdiv(n^6, d, if(ispower(d, 7), moebius(sqrtnint(d, 7))*sigma(n^6/d), 0))) \\ Altug Alkan, Oct 30 2015

a(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; f[i,1] = p^(6*f[i,2]-6)*(1+p+p^2+p^3+p^4+p^5+p^6); f[i,2] = 1;); factorback(f);} \\ Michel Marcus, Nov 12 2015

a(n) = J_7(n)/J_1(n) = J_7(n)/phi(n) = A069092(n)/A000010(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Oct 27 2010
From Álvar Ibeas, Oct 30 2015: (Start)
Multiplicative with a(p^e) = p^(6e-6) * (p^7-1) / (p-1).
For squarefree n, a(n) = A000203(n^6). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^7, where c = (1/7) * Product_{p prime} (1 + (p^6-1)/((p-1)*p^7)) = 0.2761554804... .
Sum_{k>=1} 1/a(k) = zeta(6)*zeta(7) * Product_{p prime} (1 - 2/p^7 + 1/p^13) = 1.008982290854... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^7). - Ridouane Oudra, Apr 01 2025

Definition corrected by Enrique Pérez Herrero, Oct 27 2010

A308637 Decimal expansion of Pi^3/Zeta(3).

Original entry on oeis.org

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

Offset: 2

Seiichi Manyama, Aug 23 2019

Index entries for zeta function.

-----+---------------------------------
   n |  Zeta(n)
-----+---------------------------------
   2 |     Pi^2  / 6         = A013661.
   3 |     Pi^3  / 25.79...  = A002117.
   4 |     Pi^4  / 90        = A013662.
   5 |     Pi^5  / A309926   = A013663.
   6 |     Pi^6  / 945       = A013664.
   7 |     Pi^7  / A309927   = A013665.
   8 |     Pi^8  / 9450      = A013666.
   9 |     Pi^9  / A309928   = A013667.
  10 |     Pi^10 / 93555     = A013668.
  11 |     Pi^11 / A309929   = A013669.
  12 | 691*Pi^12 / 638512875 = A013670.
...
Cf. A002432, A091925, A276120 (Zeta(3)/Pi^3).

RealDigits[Pi^3/Zeta[3], 10, 100][[1]] (* Amiram Eldar, Aug 24 2019 *)

PARI
```
Pi^3/zeta(3)
```

Pi^3/Zeta(3) = A091925/A002117.

More terms from Amiram Eldar, Aug 24 2019

A372963 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^2.

Original entry on oeis.org

1, 61, 721, 3901, 15601, 43981, 117601, 249661, 525601, 951661, 1771441, 2812621, 4826641, 7173661, 11248321, 15978301, 24137281, 32061661, 47045521, 60859501, 84790321, 108057901, 148035361, 180005581, 243765601, 294425101, 383163121, 458761501, 594822481, 686147581

Offset: 1

Seiichi Manyama, May 18 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A068963, A084218, A372962.
Cf. A084220.
Cf. A013663, A013665.

f[p_, e_] := (p^(6*e+6) - p^(6*e+2) + p^2 - 1)/(p^6-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^2*sigma(d, 6));

a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_6(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(6*e+6) - p^(6*e+2) + p^2 - 1)/(p^6-1).
Dirichlet g.f.: zeta(s)*zeta(s-6)/zeta(s-2).
Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = zeta(7)/zeta(5) = 0.972439277... . (End)
a(n) = Sum_{d|n} phi(n/d) * (n/d)^2 * sigma_6(d^2)/sigma_3(d^2). - Seiichi Manyama, May 24 2024
a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( gcd(x_1, x_2, n)/gcd(x_1, x_2, x_3, x_4, n) )^4. - Seiichi Manyama, May 25 2024

A023875 Expansion of Product_{k>=1} (1 - x^k)^(-k^6).

Original entry on oeis.org

1, 1, 65, 794, 6970, 69251, 689896, 6309849, 55654858, 483526120, 4104495070, 33968248260, 275366110929, 2192975727284, 17169583920204, 132264358228507, 1003715206329332, 7511468689508580, 55479733165442038, 404709688656248024, 2917717129031507178

Offset: 0

Olivier Gérard

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1771 (first 451 terms from Alois P. Heinz)
G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
Vaclav Kotesovec, Graph - The asymptotic ratio for 10000 terms
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 21.

Column k=6 of A144048.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^6: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*d^6, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 02 2012

max = 20; Series[ Product[1/(1 - x^k)^k^6, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x] & (* Jean-François Alcover, Mar 05 2013 *)

m=30; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^6)) \\ G. C. Greubel, Oct 31 2018

a(n) ~ exp(Pi * 2^(27/8) * n^(7/8) / (7*15^(1/8)) - 45*Zeta(7) / (8*Pi^6)) / (2^(29/16) * 15^(1/16) * n^(9/16)), where Zeta(7) = A013665 = 1.00834927738192... . - Vaclav Kotesovec, Feb 27 2015
G.f.: exp( Sum_{n>=1} sigma_7(n)*x^n/n ). - Seiichi Manyama, Mar 05 2017
a(n) = (1/n)*Sum_{k=1..n} sigma_7(k)*a(n-k). - Seiichi Manyama, Mar 05 2017

Definition corrected by Franklin T. Adams-Watters and R. J. Mathar, Dec 04 2006

A371878 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} n/gcd(x_1, x_2, x_3, x_4, x_5, n).

Original entry on oeis.org

1, 63, 727, 4031, 15621, 45801, 117643, 257983, 529981, 984123, 1771551, 2930537, 4826797, 7411509, 11356467, 16510911, 24137553, 33388803, 47045863, 62968251, 85526461, 111607713, 148035867, 187553641, 244078121, 304088211, 386356147, 474218933, 594823293, 715457421

Offset: 1

Seiichi Manyama, May 25 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Column k=5 of A372968.
Cf. A371491, A373007, A373103, A373105.
Cf. A013664, A013665.

f[p_, e_] := (p^(6*e+6) - p^(6*e+1) + p - 1)/(p^6-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, May 25 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*n/d*sigma(d, 6));

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( gcd(x_1, x_2, x_3, x_4, n)/gcd(x_1, x_2, x_3, x_4, x_5, n) )^5.
a(n) = Sum_{d|n} mu(n/d) * (n/d) * sigma_6(d).
From Amiram Eldar, May 25 2024: (Start)
Multiplicative with a(p^e) = (p^(6*e+6) - p^(6*e+1) + p - 1)/(p^6-1).
Dirichlet g.f.: zeta(s)*zeta(s-6)/zeta(s-1).
Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = zeta(7)/zeta(6) = 0.9911595361106... . (End)

A373133 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( n/gcd(x_1, x_2, x_3, n) )^3 ).

Original entry on oeis.org

1, 106, 1041, 7218, 19345, 110346, 136801, 465522, 768327, 2050570, 1947121, 7513938, 5226481, 14500906, 20138145, 29822066, 25640641, 81442662, 49651921, 139632210, 142409841, 206394826, 154751521, 484608402, 302749845, 554006986, 560366223, 987429618, 616040881

Offset: 1

Seiichi Manyama, May 26 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A059376, A084220.
Cf. A373130, A373131.
Cf. A062952, A373132, A373135.
Cf. A013662, A013665.

f[p_, e_] := (p^(6*e+4)*(p+1) - p^(3*e)*(p^4+p^3+p+1) + p^2+p)/((p^2-1)*(p^3+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2024 *)

J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
a(n, k=3, m=3) = sumdiv(n, d, J(d, k)*sigma(d^m));

a(n) = Sum_{d|n} J_3(d) * sigma(d^3), where the Jordan totient function J_3(n) = A059376(n).
From Amiram Eldar, May 26 2024: (Start)
Multiplicative with a(p^e) = (p^(6*e+4)*(p+1) - p^(3*e)*(p^4+p^3+p+1) + p^2+p)/((p^2-1)*(p^3+1)).
Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = zeta(4) * zeta(7) * Product_{p prime} (1 + 1/p^2 + 1/p^3 - 1/p^4 - 1/p^5 - 1/p^6 - 1/p^7 + 1/p^8) = 1.71945569563704656468... . (End)

A020773 Decimal expansion of 1/4.

Original entry on oeis.org

2, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Also, decimal expansion of  1/2 * integral_0^infinity 1/cosh(Pi*x) dx. - Bruno Berselli, Mar 20 2013
In the complex plane, this purely real number gives the coordinates for the inward cusp of the main cardioid of the Mandelbrot set. - Alonso del Arte, Jun 05 2016
Equals the sum of the fractional parts of the odd-indexed zeta values [Adamchik]: Sum_{k>=1} [Zeta(2k+1)-1] = 1/4 = A002117-1 + A013663-1 + A013665-1 + ... - R. J. Mathar, Jan 13 2021

V. S. Adamchi and H. M. Srivastava, Some series of the zeta and related functions, Analysis (Munich) 18 (1998) 271-288, eq (1.7)
Index entries for linear recurrences with constant coefficients, signature (1).

RealDigits[1/4, 10, 100][[1]] (* Alonso del Arte, Jun 05 2016 *)

 25 \\ Charles R Greathouse IV, Apr 15 2015

1/4 = Sum_{n >= 1} (-1)^(n+1)*n/(4*n^2-1). - Bruno Berselli, Sep 09 2020

A077454 a(n) = sigma_3(n^3)/sigma(n^3).

Original entry on oeis.org

1, 39, 511, 2359, 12621, 19929, 101179, 149943, 368089, 492219, 1611831, 1205449, 4457701, 3945981, 6449331, 9588151, 22722609, 14355471, 44576623, 29772939, 51702469, 62861409, 141611691, 76620873, 196890121, 173850339, 268218727, 238681261, 574336533, 251523909

Offset: 1

Benoit Cloitre, Nov 30 2002

			a(2) = sigma_3(2^3)/sigma(2^3) = 585/15 = 39.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A000578, A001158, A013665, A057660, A077455, A077456.

f[p_, e_] := (p^(6*e+2) + p^(3*e+1) + 1)/(p^2 + p + 1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 30] (* Amiram Eldar, Sep 09 2020 *)

PARI
```
a(n)=sumdiv(n^3,d,d^3)/sigma(n^3)
```

a(n) = my(f=factor(n^3)); sigma(f, 3)/sigma(f); \\ Michel Marcus, Sep 09 2020

a(n) = A001158(n^3)/A000203(n^3).
Multiplicative with a(p^e) = (p^(6*e+2) + p^(3*e+1) + 1)/(p^2 + p + 1). - Amiram Eldar, Sep 09 2020
Sum_{k=1..n} a(k) ~ c * n^7, where c = (zeta(7)*Pi^4/630) * Product_{p prime} (1 - 1/p^2 - 1/p^6 + 1/p^7 - 1/p^8 + 1/p^9) = 0.09343400455... . - Amiram Eldar, Oct 28 2022

A160908 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 9.

Original entry on oeis.org

1, 255, 3280, 32640, 97656, 836400, 960800, 4177920, 7173360, 24902280, 21435888, 107059200, 67977560, 245004000, 320311680, 534773760, 435984840, 1829206800, 943531280, 3187491840, 3151424000, 5466151440, 3559590240, 13703577600, 7629375000, 17334277800

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^8 such that the quotient group Z^8 / L is C_n. - Álvar Ibeas, Oct 30 2015

G. C. Greubel, Table of n, a(n) for n = 1..5000
Jin Ho Kwak and Jaeun Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
Index to Jordan function ratios J_k/J_1.

Column 8 of A263950.

Cf. A000010, A000203, A008683, A013665, A013666, A069093.

A160908[n_]:=DivisorSum[n,MoebiusMu[n/# ]*#^(9-1)/EulerPhi[n]&] (* Enrique Pérez Herrero, Oct 28 2010 *)
f[p_, e_] := p^(7*e - 7) * (p^8-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Nov 08 2022 *)

vector(30, n, sumdiv(n^7, d, if(ispower(d, 8), moebius(sqrtnint(d, 8))*sigma(n^7/d), 0))) \\ Altug Alkan, Oct 30 2015

a(n) = {f = factor(n); for (i=1, #f~, p = f[i,1]; f[i,1] = p^(7*f[i,2]-7)*(p^8-1)/(p-1); f[i,2] = 1;); factorback(f);} \\ Michel Marcus, Nov 12 2015

a(n) = J_8(n)/J_1(n) = J_8(n)/phi(n) = A069093(n)/A000010(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Oct 28 2010
From Álvar Ibeas, Oct 30 2015: (Start)
Multiplicative with a(p^e) = p^(7e-7) * (p^8-1) / (p-1).
For squarefree n, a(n) = A000203(n^7). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^8, where c = (1/8) * Product_{p prime} (1 + (p^7-1)/((p-1)*p^8)) = 0.2423008904... .
Sum_{k>=1} 1/a(k) = zeta(7)*zeta(8) * Product_{p prime} (1 - 2/p^8 + 1/p^15) = 1.004270064601... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^8). - Ridouane Oudra, Apr 01 2025

Definition corrected by Enrique Pérez Herrero, Oct 28 2010

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).

cp's OEIS Frontend

A160897 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 8.

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A308637 Decimal expansion of Pi^3/Zeta(3).

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A372963 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^2.

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A023875 Expansion of Product_{k>=1} (1 - x^k)^(-k^6).

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A371878 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} n/gcd(x_1, x_2, x_3, x_4, x_5, n).

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A373133 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( n/gcd(x_1, x_2, x_3, n) )^3 ).

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A020773 Decimal expansion of 1/4.

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A077454 a(n) = sigma_3(n^3)/sigma(n^3).