A344480 a(n) = Sum_{d|n} d * sigma_d(d), where sigma_k(n) is the sum of the k-th powers of the divisors of n.

1, 11, 85, 1103, 15631, 284795, 5764809, 134745175, 3486961642, 100097682141, 3138428376733, 107019534806039, 3937376385699303, 155577590686826319, 6568408813691811835, 295152408847835466855, 14063084452067724991027, 708238048886862707907062, 37589973457545958193355621, 2097154000001929438984022793

Offset: 1

Author

Wesley Ivan Hurt, May 20 2021

Keywords

• nonn

Comments

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

Inverse Möbius transform of n * sigma_n(n). - Wesley Ivan Hurt, Mar 31 2025

Examples

			a(6) = Sum_{d|6} d * sigma_d(d) = 1*(1^1) + 2*(1^2 + 2^2) + 3*(1^3 + 3^3) + 6*(1^6 + 2^6 + 3^6 + 6^6) = 284795.
		

Crossrefs

Cf. A245466, A321141, A334874, A343781, A344434.

Programs

Mathematica

Table[Sum[k*DivisorSigma[k, k] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 30}]

PARI

a(n) = sumdiv(n, d, d*sigma(d, d)); \\ Michel Marcus, May 21 2021