A344047 -id:A344047 - cp's OEIS frontend

A344044 a(n) = Sum_{d|n} sigma(d)^3.

1, 28, 65, 371, 217, 1820, 513, 3746, 2262, 6076, 1729, 24115, 2745, 14364, 14105, 33537, 5833, 63336, 8001, 80507, 33345, 48412, 13825, 243490, 30008, 76860, 66262, 190323, 27001, 394940, 32769, 283584, 112385, 163324, 111321, 839202, 54873, 224028, 178425, 812882, 74089

Offset: 1

Seiichi Manyama, May 08 2021

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

Cf. A002117, A007429, A065018, A344043, A344047.

a[n_] := DivisorSum[n, DivisorSigma[1, #]^3 &]; Array[a, 41] (* Amiram Eldar, May 08 2021 *)

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

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k)^3*x^k/(1-x^k)))

G.f.: Sum_{k >= 1} sigma(k)^3 * x^k/(1 - x^k).
If p is prime, a(p) = 1 + (p+1)^3.
Sum_{k=1..n} a(k) ~ c * n^4, where c = (Pi^10*zeta(3)/194400) * Product_{p prime} (1 + 2/p^2 + 2/p^3 + 1/p^5) = 1.6422194986... . - Amiram Eldar, Nov 20 2022

A344060 a(n) = Sum_{d|n} sigma(d)^n.

Original entry on oeis.org

1, 10, 65, 2483, 7777, 2990810, 2097153, 2568661988, 10604761518, 3570527751850, 743008370689, 232227195048256531, 793714773254145, 21035724521219881850, 504857283427304833025, 727429690188773950335429, 2185911559738696531969, 43567528891100073055151954340, 5242880000000000000000001

Offset: 1

Seiichi Manyama, May 08 2021

nonn

Cf. A007429, A023887, A065018, A342471, A344044, A344047, A344061.

a[n_] := DivisorSum[n, DivisorSigma[1 , #]^n &]; Array[a, 19] (* Amiram Eldar, May 08 2021 *)

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

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (sigma(k)*x)^k/(1-(sigma(k)*x)^k)))

G.f.: Sum_{k >= 1} (sigma(k) * x)^k/(1 - (sigma(k) * x)^k).

If p is prime, a(p) = 1 + (p+1)^p.

A344081 a(n) = Sum_{d|n} tau(d)^d, where tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 5, 9, 86, 33, 4109, 129, 65622, 19692, 1048613, 2049, 2176786526, 8193, 268435589, 1073741865, 152587956247, 131073, 101559956692208, 524289, 3656158441111670, 4398046511241, 17592186046469, 8388609, 4722366482871822065758

Offset: 1

Seiichi Manyama, May 09 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..719

Cf. A007425, A062367, A097988, A279789, A344047, A344080.

a[n_] := DivisorSum[n, DivisorSigma[0, #]^# &]; Array[a, 24] (* Amiram Eldar, May 09 2021 *)

PARI
```
a(n) = sumdiv(n, d, numdiv(d)^d);
```

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (numdiv(k)*x)^k/(1-x^k)))

G.f.: Sum_{k >= 1} (tau(k) * x)^k/(1 - x^k).

If p is prime, a(p) = 1 + 2^p.

A359054 a(n) = Sum_{d|n} sigma_d(d)^d.

Original entry on oeis.org

1, 26, 21953, 5554571867, 298500366308609377, 11413459460309090640625021978, 256925761343390078522337875137209684721665, 6476754651706496208416137876625690606552226172163824554588

Offset: 1

Seiichi Manyama, Dec 14 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..26

Cf. A344047, A359052, A359053.

a[n_] := DivisorSum[n, DivisorSigma[#, #]^# &]; Array[a, 8] (* Amiram Eldar, Aug 27 2023 *)

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

my(N=10, x='x+O('x^N)); Vec(sum(k=1, N, (sigma(k, k)*x)^k/(1-x^k)))

G.f.: Sum_{k >= 1} (sigma_k(k) * x)^k/(1 - x^k).

cp's OEIS Frontend

A344044 a(n) = Sum_{d|n} sigma(d)^3.

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A344060 a(n) = Sum_{d|n} sigma(d)^n.

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A344081 a(n) = Sum_{d|n} tau(d)^d, where tau(n) is the number of divisors of n.

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Formula

A359054 a(n) = Sum_{d|n} sigma_d(d)^d.

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