A344060 -id:A344060 - cp's OEIS frontend

A344080 a(n) = Sum_{d|n} tau(d)^n, where tau(n) is the number of divisors of n.

1, 5, 9, 98, 33, 4225, 129, 72354, 20196, 1050625, 2049, 2194099186, 8193, 268468225, 1073807361, 156925970179, 131073, 101629064089930, 524289, 3657261440572306, 4398050705409, 17592194433025, 8388609, 4727105427440383342818, 847322163876, 4503599761588225

Offset: 1

Seiichi Manyama, May 09 2021

nonn

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

Cf. A007425, A062367, A097988, A279789, A344060, A344081.

a[n_] := DivisorSum[n, DivisorSigma[0, #]^n &]; Array[a, 26] (* Amiram Eldar, May 09 2021 *)

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

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

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

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

A359052 a(n) = Sum_{d|n} sigma_d(d)^n.

Original entry on oeis.org

1, 26, 21953, 5554572467, 298500366308609377, 11413459460309090641106905930, 256925761343390078522337875137209684721665, 6476754651706496208416137876625690606583079440495100502628

Offset: 1

Seiichi Manyama, Dec 14 2022

nonn

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

Cf. A344060, A359053, A359054.

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

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

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

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

A344061 a(n) = Sum_{d|n} sigma(d)^(n/d).

Original entry on oeis.org

1, 4, 5, 17, 7, 56, 9, 146, 78, 298, 13, 1501, 15, 2276, 1265, 9219, 19, 25716, 21, 77519, 16929, 177328, 25, 739582, 7808, 1594562, 264382, 5611241, 31, 15699452, 33, 48863172, 4196081, 129140542, 312753, 447589422, 39, 1162261928, 67111665, 3771805472, 43, 10764897556, 45

Offset: 1

Seiichi Manyama, May 08 2021

nonn

Cf. A007429, A055225, A279789, A309369, A342471, A344060.

a[n_] := DivisorSum[n, DivisorSigma[1 , #]^(n/#) &]; Array[a, 43] (* Amiram Eldar, May 08 2021 *)

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

my(N=66, 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) = 2 + p.

cp's OEIS Frontend

A344080 a(n) = Sum_{d|n} tau(d)^n, where tau(n) is the number of divisors of n.

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A359052 a(n) = Sum_{d|n} sigma_d(d)^n.

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Mathematica

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Formula

A344061 a(n) = Sum_{d|n} sigma(d)^(n/d).

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