A013957 -id:A013957 - cp's OEIS frontend

A362870 a(n) = sigma_29(n), the sum of the 29th powers of the divisors of n.

1, 536870913, 68630377364884, 288230376688582657, 186264514923095703126, 36845653355419807219092, 3219905755813179726837608, 154742505198902911050973185, 4710128697246313465298968573, 100000000186264514923632574038, 1586309297171491574414436704892

Offset: 1

Vaclav Kotesovec, May 07 2023

In general, for k > 0, Sum_{n>=1} sigma_(4*k+1)(n) / exp(2*Pi*n) = Bernoulli(4*k+2)/(8*k+4). For k = 0, Sum_{n>=1} sigma(n)/exp(2*Pi*n) = 1/24 - 1/(8*Pi) = Bernoulli(2)/4 - 1/(8*Pi).

This formula can best be understood as a statement about the divided Bernoulli numbers b(n) = B(n) / n. Then you can say: If v is twice an odd number greater than 1 (i.e., v = 4*n + 2, a term of A016825 that is greater than 2), then b(v) = 2 * Sum_{j>=1} sigma_{v - 1}(j) / exp(2*Pi*j) = A358625(v) / A075180(v - 1). - Peter Luschny, May 08 2023

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Peter Luschny, Is this a new representation of (some) Bernoulli numbers?, Mathematics Stack Exchange.
Index entries for sequences related to sigma(n).

Cf. A000203 (sigma_1), A001160 (sigma_5), A013957 (sigma_9), A013961 (sigma_13), A013965 (sigma_17), A013969 (sigma_21), A281959 (sigma_25).

Cf. A075180, A358625, A016825, A082771.

with(NumberTheory): seq(SumOfDivisors(k, 29), k = 1..20);

Mathematica
```
DivisorSigma[29, Range[20]]
```

for(n=1, 20, print1(direuler( p=2, n, 1 / (1 - X) /(1 - p^29*X))[n], ", "))

from sympy import divisor_sigma
def A362870(n): return divisor_sigma(n,29) # Chai Wah Wu, May 07 2023

G.f.: Sum_{k>=1} k^29 * x^k / (1-x^k).
Dirichlet g.f.: zeta(s-29)*zeta(s).
Sum_{k=1..n} a(k) ~ zeta(30) * n^30 / 30.
Sum_{n>=1} a(n)/exp(2*Pi*n) = 1723168255201/171864 = Bernoulli(30)/60.
Multiplicative with a(p^e) = (p^(29*e+29)-1)/(p^29-1). - Amiram Eldar, Oct 29 2023

A055703 Numbers k such that k | sigma_9(k) - phi(k)^9.

Original entry on oeis.org

1, 2, 12, 54, 76, 90, 216, 423, 514, 531, 621, 2166, 2241, 2772, 2976, 4752, 5154, 5400, 5481, 6264, 7290, 7344, 9018, 9144, 9470, 9555, 14094, 14904, 19494, 21222, 23780, 25848, 28323, 34830, 34911, 38220, 40122, 48768, 49079, 55782, 59400

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

sigma_9(k) is the sum of the 9th powers of the divisors of k (A013957).

Amiram Eldar, Table of n, a(n) for n = 1..250

Cf. A000010, A013957.

Do[If[Mod[DivisorSigma[9, n]-EulerPhi[n]^9, n]==0, Print[n]], {n, 1, 10^5}]

isok(n) = !((sigma(n, 9) - eulerphi(n)^9) % n); \\ Michel Marcus, Mar 02 2014

Definition corrected by Michel Marcus, Mar 02 2014

A081866 a(n) = sigma_9(2n-1).

Original entry on oeis.org

1, 19684, 1953126, 40353608, 387440173, 2357947692, 10604499374, 38445332184, 118587876498, 322687697780, 794320419872, 1801152661464, 3814699218751, 7625984925160, 14507145975870, 26439622160672, 46413842369328, 78815680978608, 129961739795078, 208738965677816

Offset: 1

Benoit Cloitre, Apr 12 2003

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A013957.

DivisorSigma[9,Range[1,41,2]] (* Harvey P. Dale, Dec 01 2013 *)

a(n) = sigma(2*n-1, 9); \\ Amiram Eldar, Jan 08 2025

Sum_{k=1..n} a(k) ~ c * n^10, where c = 31 * Pi^10 / 56700 = 51.200872... . - Amiram Eldar, Jan 08 2025

A279926 a(n) = Sum_{k=1..n-1} sigma_3(k)*sigma_9(n-k).

Original entry on oeis.org

0, 1, 522, 24329, 454250, 4905766, 36532244, 207705929, 961214238, 3784166376, 13066960126, 40511160326, 114681233758, 300599979884, 737035375772, 1705830324553, 3751239987240, 7887626314003, 15927815870322, 31031953887704, 58508991327728, 107133058597170

Offset: 1

Seiichi Manyama, Dec 23 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A001158, A013957, A013961, A279888, A279917.

Table[Sum[If[k == 0, 0, DivisorSigma[3, k]] DivisorSigma[9, n - k], {k, 0, n - 1}], {n, 22}] (* Michael De Vlieger, Dec 23 2016 *)

a(n) = sum(k=1, n-1, sigma(k, 3)*sigma(n-k, 9)) \\ Felix Fröhlich, Dec 23 2016

a(n) = (sigma_13(n) - 11*sigma_9(n) + 10*sigma_3(n))/2640.

A280021 Expansion of phi_{11, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 2052, 177156, 4202512, 48828150, 363524112, 1977326792, 8606744640, 31382654013, 100195363800, 285311670732, 744500215872, 1792160394206, 4057474577184, 8650199741400, 17626613022976, 34271896307922, 64397206034676, 116490258898580, 205200886312800

Offset: 0

Seiichi Manyama, Feb 22 2017

Multiplicative because A013957 is. - Andrew Howroyd, Jul 23 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A282097 (phi_{3, 2}), A282099 (phi_{5, 2}), A282751 (phi_{7, 2}), A282753 (phi_{9, 2}), this sequence (phi_{11, 2}).
Cf. A282549 (E_2*E_4^3), A282792 (E_2^2*E_4*E_6), A282576 (E_2*E_6^2), A058550 (E_4^2*E_6 = E_14).
Cf. A013957 (sigma_9(n)), A282254 (n*sigma_9(n)), this sequence (n^2*sigma_9(n)).
Cf. A013668 (zeta(10)).

Table[If[n>0, n^2 * DivisorSigma[9, n], 0], {n, 0, 20}] (* Indranil Ghosh, Mar 12 2017 *)

for(n=0, 20, print1(if(n==0, 0, n^2 * sigma(n, 9)),", ")) \\ Indranil Ghosh, Mar 12 2017

a(n) = n^2*A013957(n) for n > 0.
a(n) = (6*A282549(n) - 5*A282792(n) + 4*A282576(n) - 5*A058550(n))/1728.
Sum_{k=1..n} a(k) ~ zeta(10) * n^12 / 12. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^(2*e) * (p^(9*e+9)-1)/(p^9-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-11). (End)

cp's OEIS Frontend

A362870 a(n) = sigma_29(n), the sum of the 29th powers of the divisors of n.

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A055703 Numbers k such that k | sigma_9(k) - phi(k)^9.

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A081866 a(n) = sigma_9(2n-1).

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A279926 a(n) = Sum_{k=1..n-1} sigma_3(k)*sigma_9(n-k).

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A280021 Expansion of phi_{11, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

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