A069153 a(n) = Sum_{d|n} d*(d-1)/2.

0, 1, 3, 7, 10, 19, 21, 35, 39, 56, 55, 91, 78, 113, 118, 155, 136, 208, 171, 252, 234, 287, 253, 395, 310, 404, 390, 497, 406, 614, 465, 651, 586, 698, 626, 910, 666, 875, 822, 1060, 820, 1202, 903, 1239, 1144, 1289, 1081, 1643, 1197, 1581, 1414, 1736

Offset: 1

Author

Benoit Cloitre, Apr 08 2002

Keywords

• easy
• nonn

Comments

Inverse Mobius transform of A000217. - R. J. Mathar, Jan 19 2009

Examples

			x^2 + 3*x^3 + 7*x^4 + 10*x^5 + 19*x^6 + 21*x^7 + 35*x^8 + 39*x^9 + 56*x^10 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Crossrefs

Cf. A000203, A001157, A002117, A007437, A086666, A134840.

Cf. A032741, A065608, A363604, A363605, A363606.

Programs

Maple

with(numtheory):
seq((1/2)*(sigma[2](n) - sigma[1](n)), n = 1..100); # Peter Bala, Jan 21 2021

Mathematica

A069153[n_]:=Plus@@Binomial[Divisors[n],2];Array[A069153,100] (* Enrique Pérez Herrero, Feb 21 2012 *)

PARI

{a(n) = if( n<1, 0, sumdiv(n, d, d^2 - d) / 2)}

PARI

a(n) = my(f = factor(n)); (sigma(f, 2) - sigma(f)) / 2; \\ Amiram Eldar, Jan 01 2025

Formula

G.f.: Sum_{k>0} x^(2*k)/(1-x^k)^3. - Vladeta Jovovic, Dec 17 2002

Row sums of triangle A134840. - Gary W. Adamson, Nov 12 2007

G.f. A(x) = (1/2) * x * d/dx log( B(x) ) where B() is g.f. for A052847. - Michael Somos, Feb 12 2008

G.f.: Sum_{k>0} ((k^2 - k) / 2) * x^k / (1 - x^k). - Michael Somos, Feb 12 2008

From Peter Bala, Jan 21 2021: (Start)

a(n) = (1/2)*(sigma_2(n) - sigma_1(n)) = (1/2)*(A001157(n) A000203(n)) = (1/2)*A086666.

G.f.: A(x) = (1/2)* Sum_{n >= 1} x^(n^2)*( n*(n-1)*x^(3*n) - (n^2 + n - 2)*x^(2*n) + n*(3 - n)*x^n + n*(n - 1) )/(1 - x^n)^3. - differentiate equation 5 in Arndt twice w.r.t x and set x = 1. (End)

From Amiram Eldar, Jan 01 2025: (Start)

Dirichlet g.f.: zeta(s) * (zeta(s-2) - zeta(s-1)) / 2.

Sum_{k=1..n} a(k) ~ (zeta(3)/6) * n^3. (End)