A211780 a(n) = Sum_{d|n, d<n} d * tau(n / d), where tau = A000005 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

1,2

Comments

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

Links

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

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A029632 A089588 A325211 * A014840 A218756 A317329

Adjacent sequences: A211777 A211778 A211779 * A211781 A211782 A211783

Keyword

nonn

Author

Jaroslav Krizek, Apr 20 2012

Extensions

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

Status

approved