A211780 a(n) = Sum_{d|n, dA000005 is the number of divisors.

0, 2, 2, 7, 2, 14, 2, 18, 9, 18, 2, 43, 2, 22, 20, 41, 2, 54, 2, 57, 24, 30, 2, 106, 13, 34, 31, 71, 2, 110, 2, 88, 32, 42, 28, 162, 2, 46, 36, 142, 2, 138, 2, 99, 81, 54, 2, 237, 17, 102, 44, 113, 2, 178, 36, 178, 48, 66, 2, 325, 2, 70, 99, 183, 40, 194, 2

Offset: 1

Author

Jaroslav Krizek, Apr 20 2012

Keywords

• nonn

Comments

Numbers n such that n divides a(n) are given in A068978.

Examples

			For n = 12: Sum_{d|n, d<n} d * tau(n / d) = 1*6 + 2*4 + 3*3 + 4*2 + 6*2 = 43.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..27144 (first 1000 terms from Jaroslav Krizek)

Crossrefs

Cf. A000005, A000203, A007429, A068978, A098198, A211779.

Programs

Mathematica

Table[Sum[d*DivisorSigma[0, n/d], {d, Most[Divisors[n]]}], {n, 100}] (* T. D. Noe, Apr 27 2012 *)

PARI

A211780(n) = sumdiv(n, d, sigma(d))-n; \\ Antti Karttunen, Nov 13 2017

Python

A211780=lambda n:sum(sigma(d) for d in divisors(n, generator=True))-n
from sympy import divisor_sigma as sigma, divisors # M. F. Hasler, Jun 03 2024

Formula

a(n) = A007429(n) - n = A211779(n) + A000203(n) - n .

a(n) = (Sum_{d|n} A000203(d)) - n. - Antti Karttunen, Nov 13 2017

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Pi^4/36 - 1 = 1.705808... . - Amiram Eldar, Jun 06 2024

Extensions

Name edited by M. F. Hasler, Jun 03 2024