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

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

Author

Wesley Ivan Hurt, Jun 12 2021

Keywords

• nonn

Comments

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

Examples

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

Links

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

Crossrefs

Cf. A000005 (tau), A174937, A345271.

Programs

Mathematica

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

PARI

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

Formula

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