A345271 -id:A345271 - cp's OEIS frontend

A174937 a(n) = Sum_{d|n} d^tau(d).

1, 5, 10, 69, 26, 1310, 50, 4165, 739, 10030, 122, 2987358, 170, 38470, 50660, 1052741, 290, 34014263, 362, 64010094, 194540, 234382, 530, 110078305630, 15651, 457150, 532180, 481928838, 842, 656100061960

Offset: 1

Jaroslav Krizek and Paul D. Hanna, Apr 02 2010

nonn

Here tau(n) = A000005(n) = the number of divisors of n.

			For n = 4, A007955(n) = b(n): a(4) = b(1)^2 + b(2)^2 + b(4)^2 = 1^2 + 2^2 + 8^2 = 69.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005 (tau), A174472, A174473, A345270, A345271.

a[n_] := DivisorSum[n, #^DivisorSigma[0, #] &]; Array[a, 30] (* Amiram Eldar, Oct 08 2021 *)

PARI
```
{a(n)=sumdiv(n,d,d^sigma(d,0))}
```

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k^numdiv(k)*x^k/(1-x^k))) \\ Seiichi Manyama, Oct 14 2021

Also a(n) = Sum_{d|n} A007955(d)^2, where A007955(m) = product of divisors of m.
Logarithmic derivative of A174473.
G.f.: Sum_{k>=1} k^tau(k) * x^k/(1 - x^k). - Seiichi Manyama, Oct 14 2021

A345270 a(n) = Sum_{d|n} d^tau(n/d).

Original entry on oeis.org

1, 3, 4, 9, 6, 20, 8, 33, 19, 40, 12, 108, 14, 68, 50, 161, 18, 225, 20, 278, 80, 148, 24, 850, 51, 200, 136, 600, 30, 1114, 32, 1089, 164, 328, 110, 3387, 38, 404, 218, 2450, 42, 3214, 44, 1892, 558, 580, 48, 12596, 99, 1409, 350, 2958, 54, 8630, 202, 6370, 428, 904, 60, 33042

Offset: 1

Wesley Ivan Hurt, Jun 12 2021

nonn

If p is a prime, a(p) = Sum_{d|p} d^tau(p/d) = 1^2 + p^1 = p + 1.

			a(10) = Sum_{d|10} d^tau(10/d) = 1^4 + 2^2 + 5^2 + 10^1 = 40.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005 (tau), A174937, A345271.

Table[Sum[k^DivisorSigma[0, n/k] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 60}]

a(n) = sumdiv(n, d, d^numdiv(n/d)); \\ Michel Marcus, Oct 08 2021

a(n) = Sum_{d|n} (n/d)^tau(d). - Wesley Ivan Hurt, Jun 09 2023

A345895 a(n) = Sum_{d|n} n^sigma(d).

Original entry on oeis.org

1, 10, 84, 16452, 15630, 2176783854, 5764808, 35184374186504, 2541865834899, 1000000000001001010, 3138428376732, 1648446623609512552867179993036, 3937376385699302, 3214199700417740938226879430, 16834112196028232574474331890, 21267647932558653967613834469360799760

Offset: 1

Wesley Ivan Hurt, Jun 28 2021

nonn

			a(4) = Sum_{d|4} 4^sigma(d) = 4^1 + 4^3 + 4^7 = 16452.

Seiichi Manyama, Table of n, a(n) for n = 1..167

Cf. A000203 (sigma), A104128, A174472, A345271.

a[n_] := DivisorSum[n, n^DivisorSigma[1, #] &]; Array[a, 16] (* Amiram Eldar, Jun 29 2021 *)

a(n) = sumdiv(n, d, n^sigma(d)); \\ Michel Marcus, Jun 29 2021

a(prime(n)) = A104128(n). - Michel Marcus, Jun 29 2021

cp's OEIS Frontend

A174937 a(n) = Sum_{d|n} d^tau(d).

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A345270 a(n) = Sum_{d|n} d^tau(n/d).

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A345895 a(n) = Sum_{d|n} n^sigma(d).

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