A013961 -id:A013961 - 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

A055707 Numbers k such that k | sigma_13(k) - phi(k)^13.

Original entry on oeis.org

1, 2, 12, 34, 42, 90, 170, 198, 402, 434, 456, 482, 494, 2046, 4086, 4518, 7520, 7605, 8622, 9632, 10924, 28280, 51570, 51714, 74124, 77724, 100172, 139653, 143136, 176760, 294588, 399980, 471826, 675356, 690534, 1358360, 1577696, 2089074, 2121940, 2136256

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

sigma_13(k) is the sum of the 13th powers of the divisors of k (A013961).

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000010, A013961.

Do[If[Mod[DivisorSigma[13, n]-EulerPhi[n]^13, n]==0, Print[n]], {n, 1, 10^5}]

isok(n) = !((sigma(n, 13) - eulerphi(n)^13) % n); \\ Michel Marcus, Mar 02 2014

Definition corrected and more terms from Michel Marcus, Mar 02 2014

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.

A158033 a(n) = sigma_(Fibonacci(n)) (n).

Original entry on oeis.org

1, 3, 10, 73, 3126, 1686434, 96889010408, 9223376434903384065, 278128389443693527934467475898331, 10000000000000000277555756156289135105943945819724042094

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 11 2009

nonn

Cf. A000045, A000203, A001157, A001158, A001160, A013956, A013961, A013969.

a:= n-> numtheory[sigma][combinat[fibonacci](n)](n):
seq(a(n), n=1..10);  # Alois P. Heinz, Feb 10 2020

Table[DivisorSigma[Fibonacci[n],n],{n,10}] (* Harvey P. Dale, Nov 24 2013 *)

a(n) = sigma(n, fibonacci(n)); \\ Michel Marcus, Feb 09 2020

[sigma(n,fibonacci(n))for n in range(1,11)] # Zerinvary Lajos, Jun 04 2009

Name edited by Michel Marcus, Feb 09 2020

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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A055707 Numbers k such that k | sigma_13(k) - phi(k)^13.

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

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A158033 a(n) = sigma_(Fibonacci(n)) (n).

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