A345895 -id:A345895 - cp's OEIS frontend

A174472 a(n) = Sum_{d|n} d^sigma(d).

1, 9, 82, 16393, 15626, 2176782426, 5764802, 35184372105225, 2541865828411, 1000000000000015634, 3138428376722, 1648446623609512543953220489306, 3937376385699290, 3214199700417740936756852426, 16834112196028232574462906332, 21267647932558653966460948148857618441

Offset: 1

Paul D. Hanna, Apr 04 2010

nonn

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

Cf. A000203 (sigma), A062796, A174471, A174937, A345895.

a[n_] := DivisorSum[n, #^DivisorSigma[1, #] &]; Array[a, 16] (* Amiram Eldar, Oct 08 2021 *)

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

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^sigma(k)*x^k/(1-x^k))) \\ Seiichi Manyama, Oct 14 2021

Logarithmic derivative of A174471.

G.f.: Sum_{k>=1} k^sigma(k) * x^k/(1 - x^k). - Seiichi Manyama, Oct 14 2021

A345271 a(n) = Sum_{d|n} n^tau(d).

Original entry on oeis.org

1, 6, 12, 84, 30, 1374, 56, 4680, 819, 10210, 132, 3008748, 182, 38822, 51090, 1118480, 306, 34123698, 380, 64168820, 195384, 235246, 552, 110267095704, 16275, 458354, 551880, 482528508, 870, 656102432730, 992, 1108378656, 1188132, 1338682, 1503110, 101564312008644

Offset: 1

Wesley Ivan Hurt, Jun 12 2021

nonn

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

			a(10) = Sum_{d|10} 10^tau(d) = 10^1 + 10^2 + 10^2 + 10^4 = 10210.

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

Cf. A000005 (tau), A174937, A345270, A345895.

Table[Sum[n^DivisorSigma[0, k] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 50}]

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

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A174472 a(n) = Sum_{d|n} d^sigma(d).

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A345271 a(n) = Sum_{d|n} n^tau(d).

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