A013668 -id:A013668 - cp's OEIS frontend

A282254 Coefficients in q-expansion of (3E_4^3 + 2E_6^2 - 5E_2E_4*E_6)/1584, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

0, 1, 1026, 59052, 1050628, 9765630, 60587352, 282475256, 1075843080, 3486961557, 10019536380, 25937424612, 62041684656, 137858491862, 289819612656, 576679982760, 1101663313936, 2015993900466, 3577622557482, 6131066257820, 10260044315640

Offset: 0

Seiichi Manyama, Feb 10 2017

Multiplicative because A013957 is. - Andrew Howroyd, Jul 25 2018

D. H. Lehmer shows that a(n) == tau(n) (mod 7) for n > 0, where tau is Ramanujan's tau function (A000594). Furthermore, if n == 3, 5, 6 (mod 7) then a(n) == tau(n) (mod 49). See the Wikipedia link below. - Jianing Song, Aug 12 2020

			a(6) = 1^10*6^1 + 2^10*3^1 + 3^10*2^1 + 6^10*1^1 = 60587352.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Wikipedia, Congruences for the tau function.

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), A282060 (phi_{8, 1}), this sequence (phi_{10, 1}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008411 (E_4^3), A280869 (E_6^2), A282102 (E_2*E_4*E_6).
Cf. A013957, A013668, A196292.

Table[If[n>0, n * DivisorSigma[9, n], 0], {n, 0, 20}] (* Indranil Ghosh, Mar 12 2017 *)

for(n=0, 20, print1(if(n==0, 0, n * sigma(n, 9)),", ")) \\ Indranil Ghosh, Mar 12 2017

G.f.: phi_{10, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = (3*A008411(n) + 2*A280869(n) - 5*A282102(n))/1584.
If p is a prime, a(p) = p^10 + p = A196292(p).
a(n) = n*A013957(n) for n > 0, where A013957(n) is sigma_9(n), the sum of the 9th powers of the divisors of n. - Seiichi Manyama, Feb 18 2017
Multiplicative with a(p^e) = p^e*(p^(9*(e+1))-1)/(p^9-1). - Jianing Song, Aug 12 2020
From Amiram Eldar, Oct 30 2023: (Start)
Dirichlet g.f.: zeta(s-1)*zeta(s-10).
Sum_{k=1..n} a(k) ~ zeta(10) * n^11 / 11. (End)

A321554 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^9.

Original entry on oeis.org

1, 511, 19684, 261631, 1953126, 10058524, 40353608, 133955071, 387440173, 998047386, 2357947692, 5149944604, 10604499374, 20620693688, 38445332184, 68584996351, 118587876498, 197981928403, 322687697780, 510998308506, 794320419872, 1204911270612, 1801152661464, 2636771617564, 3814699218751

Offset: 1

N. J. A. Sloane, Nov 23 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 similar sequences.

Cf. A013668.

f[p_, e_] := (p^(9*e + 9) - 1)/(p^9 - 1); f[2, e_] := (255*2^(9*e + 1) + 1)/511; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)

apply( A321554(n)=sumdiv(n, d, (-1)^(n\d-1)*d^9), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^9*x^k/(1 + x^k). - Seiichi Manyama, Nov 24 2018
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (255*2^(9*e+1)+1)/511, and a(p^e) = (p^(9*e+9) - 1)/(p^9 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^10, where c = 511*zeta(10)/5120 = 0.0999039... . (End)

A347220 Decimal expansion of Sum_{k=2..10} zeta(k).

Original entry on oeis.org

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

Offset: 1

Sean A. Irvine, Aug 24 2021

			9.9990146223805454454075219574377780810680...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A002117, A013661, A013662, A013663, A013664, A013665, A013666, A013667, A013668.

Cf. A347213, A347214, A347215, A347216, A347217, A347218, A347219.

RealDigits[Total[Zeta[Range[2, 10]]], 10, 120][[1]] (* Amiram Eldar, Jun 07 2023 *)

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

A344306 Number of cyclic subgroups of the group (C_n)^10, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 1024, 29525, 524800, 2441407, 30233600, 47079209, 268698112, 581150417, 2500000768, 2593742461, 15494720000, 11488207655, 48209110016, 72082541675, 137573433856, 125999618779, 595098027008, 340614792101, 1281250393600

Offset: 1

Seiichi Manyama, May 14 2021

Inverse Moebius transform of A160957.

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, On the number of cyclic subgroups of a finite abelian group, arXiv: 1203.6201 [math.GR], 2012.

Cf. A000010, A013668, A160957, A060648, A064969, A280184, A344219, A344302, A344303, A344304, A344305.

f[p_, e_] := 1 + ((p^10 - 1)/(p - 1))*((p^(9*e) - 1)/(p^9 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 15 2022 *)

a160957(n) = sumdiv(n, d, moebius(n/d)*d^10)/eulerphi(n);
a(n) = sumdiv(n, d, a160957(d));

a(n) = Sum_{x_1|n, x_2|n, ..., x_10|n} phi(x_1)*phi(x_2)* ... *phi(x_10)/phi(lcm(x_1, x_2, ..., x_10)).
If p is prime, a(p) = 1 + (p^10 - 1)/(p - 1).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + ((p^10 - 1)/(p - 1))*((p^(9*e) - 1)/(p^9 - 1)).
Sum_{k=1..n} a(k) ~ c * n^10, where c = (zeta(10)/10) * Product_{p prime} ((1-1/p^9)/(p^2*(1-1/p))) = 0.1944248708... . (End)

A157291 Decimal expansion of Zeta(5)/Zeta(10).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Feb 26 2009

The product_{p = primes = A000040} (1+1/p^5), the fifth-power analog to A082020.

			1.035897477277500224... = (1+1/2^5)*(1+1/3^5)*(1+1/5^5)*(1+1/7^5)*...

Cf. A013663, A013668, A050997, A082020.

Cf. A005117, A113850.

Maple
```
evalf(Zeta(5)/Zeta(10)) ;
```

RealDigits[Zeta[5]/Zeta[10],10,120][[1]] (* Harvey P. Dale, Apr 06 2013 *)

Equals A013663/A013668 = Product_{i>=1} (1+1/A050997(i)).
Equals Sum_{k>=1} 1/A005117(k)^5 = 1 + Sum_{k>=1} 1/A113850(k). - Amiram Eldar, May 22 2020
Equals 93555 * zeta(5) / Pi^10. - Vaclav Kotesovec, May 22 2020

A160957 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 11.

Original entry on oeis.org

1, 1023, 29524, 523776, 2441406, 30203052, 47079208, 268173312, 581120892, 2497558338, 2593742460, 15463962624, 11488207654, 48162029784, 72080070744, 137304735744, 125999618778, 594486672516, 340614792100, 1278749869056

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^10 such that the quotient group Z^10 / L is C_n. - Álvar Ibeas, Nov 26 2015

Álvar Ibeas, Table of n, a(n) for n = 1..10000
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 10 of A263950.

Cf. A000010, A000203, A013667, A013668, A069095.

b = 11; Table[Sum[MoebiusMu[n/d] d^(b - 1)/EulerPhi@ n, {d, Divisors@ n}], {n, 20}] (* Michael De Vlieger, Nov 27 2015 *)
f[p_, e_] := p^(9*e - 9) * (p^10-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Nov 08 2022 *)

vector(100, n, sumdiv(n^9, d, if(ispower(d, 10), moebius(sqrtnint(d, 10))*sigma(n^9/d), 0))) \\ Altug Alkan, Nov 26 2015

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^10 - 1)*f[i,1]^(9*f[i,2] - 9)/(f[i,1] - 1));} \\ Amiram Eldar, Nov 08 2022

a(n) = A069095(n)/A000010(n). - R. J. Mathar, Jul 12 2011
From Álvar Ibeas, Nov 26 2015: (Start)
Multiplicative with a(p^e) = p^(9e-9) * (p^10-1) / (p-1).
For squarefree n, a(n) = A000203(n^9). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^10, where c = (1/10) * Product_{p prime} (1 + (p^9-1)/((p-1)*p^10)) = 0.1942316928... .
Sum_{k>=1} 1/a(k) = zeta(9)*zeta(10) * Product_{p prime} (1 - 2/p^10 + 1/p^19) = 1.0010137674... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^10). - Ridouane Oudra, Apr 02 2025

Definition corrected by Enrique Pérez Herrero, Oct 30 2010

A160960 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 12.

Original entry on oeis.org

1, 2047, 88573, 2096128, 12207031, 181308931, 329554457, 2146435072, 5230147077, 24987792457, 28531167061, 185660345344, 149346699503, 674597973479, 1081213356763, 2197949513728, 2141993519227, 10706111066619, 6471681049901, 25587499475968, 29189626919861, 58403298973867

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^11 such that the quotient group Z^11 / L is C_n. - Álvar Ibeas, Nov 26 2015

Álvar Ibeas, Table of n, a(n) for n = 1..10000
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 11 of A263950.

Cf. A000010, A000203, A008683, A013668, A013669.

b = 12; Table[Sum[MoebiusMu[n/d] d^(b - 1)/EulerPhi@ n, {d, Divisors@ n}], {n, 18}] (* Michael De Vlieger, Nov 27 2015 *)
f[p_, e_] := p^(10*e - 10) * (p^11-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Nov 08 2022 *)

vector(100, n, sumdiv(n^10, d, if(ispower(d, 11), moebius(sqrtnint(d, 11))*sigma(n^10/d), 0))) \\ Altug Alkan, Nov 26 2015

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^11 - 1)*f[i,1]^(10*f[i,2] - 10)/(f[i,1] - 1));} \\ Amiram Eldar, Nov 08 2022

a(n) = J_11(n)/J_1(n) where J_11 and J_1(n) = A000010(n) are Jordan functions. - R. J. Mathar, Jul 12 2011
From Álvar Ibeas, Nov 26 2015: (Start)
Multiplicative with a(p^e) = p^(10e-10) * (p^11-1) / (p-1).
For squarefree n, a(n) = A000203(n^10). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^11, where c = (1/11) * Product_{p prime} (1 + (p^10-1)/((p-1)*p^11)) = 0.1766326404... .
Sum_{k>=1} 1/a(k) = zeta(10)*zeta(11) * Product_{p prime} (1 - 2/p^11 + 1/p^21) = 1.0005003781952... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^11). - Ridouane Oudra, Apr 02 2025

Definition corrected by Enrique Pérez Herrero, Oct 30 2010

A321813 Sum of 9th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 19684, 1, 1953126, 19684, 40353608, 1, 387440173, 1953126, 2357947692, 19684, 10604499374, 40353608, 38445332184, 1, 118587876498, 387440173, 322687697780, 1953126, 794320419872, 2357947692, 1801152661464, 19684, 3814699218751

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).
Eric Weisstein's World of Mathematics, Odd Divisor Function.
Index entries for sequences mentioned by Glaisher.

Column k=9 of A285425.
Cf. A050999, A051000, A051001, A051002, A321810 - A321816 (analog for 2nd .. 12th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Cf. A000265, A013668, A013957.

a[n_] := DivisorSum[n, #^9 &, OddQ[#] &]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)

apply( A321813(n)=sigma(n>>valuation(n,2),9), [1..30]) \\ M. F. Hasler, Nov 26 2018

from sympy import divisor_sigma
def A321813(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),9)) # Chai Wah Wu, Jul 16 2022

a(n) = A013957(A000265(n)) = sigma_9(odd part of n); in particular, a(2^k) = 1 for all k >= 0. - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (2*k - 1)^9*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 07 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 1 and a(p^e) = (p^(9*e+9)-1)/(p^9-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^10, where c = zeta(10)/20 = Pi^10/1871100 = 0.0500497... . (End)

A347330 Decimal expansion of zeta(10) / zeta(5).

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Aug 26 2021

			0.96534649609163603621377296424322122474050...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A008836, A182448, A347328, A347329, A347331.

Cf. A013663, A013668.

RealDigits[Zeta[10] / Zeta[5], 10, 120][[1]] (* Amiram Eldar, Jun 06 2023 *)

Equals Sum_{k>=1} A008836(k) / k^5.
Equals Product_{p prime} 1/(1+p^(-5)). [corrected by Amiram Eldar, Jun 06 2023]
Equals 1/A157291. - R. J. Mathar, Jul 20 2025

cp's OEIS Frontend

A282254 Coefficients in q-expansion of (3*E_4^3 + 2*E_6^2 - 5*E_2*E_4*E_6)/1584, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A321554 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^9.

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A347220 Decimal expansion of Sum_{k=2..10} zeta(k).

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

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A344306 Number of cyclic subgroups of the group (C_n)^10, where C_n is the cyclic group of order n.

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A157291 Decimal expansion of Zeta(5)/Zeta(10).

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A160957 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 11.

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A160960 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 12.

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A282254 Coefficients in q-expansion of (3E_4^3 + 2E_6^2 - 5E_2E_4*E_6)/1584, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.