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A211780
a(n) = Sum_{d|n, d<n} d * tau(n / d), where tau = A000005 is the number of divisors.
3
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
KEYWORD
nonn
AUTHOR
Jaroslav Krizek, Apr 20 2012
EXTENSIONS
Name edited by M. F. Hasler, Jun 03 2024
STATUS
approved