A086666 a(n) = sigma_2(n) - sigma_1(n).

0, 2, 6, 14, 20, 38, 42, 70, 78, 112, 110, 182, 156, 226, 236, 310, 272, 416, 342, 504, 468, 574, 506, 790, 620, 808, 780, 994, 812, 1228, 930, 1302, 1172, 1396, 1252, 1820, 1332, 1750, 1644, 2120, 1640, 2404, 1806, 2478, 2288, 2578, 2162, 3286, 2394, 3162

Offset: 1

Author

Jon Perry, Jul 27 2003

Keywords

• nonn
• easy

Comments

Total area of all distinct L X W rectangles such that s + t = n, 1 <= s <= t, s | n, L = n/s and W = t/s. - Wesley Ivan Hurt, Aug 01 2025

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Crossrefs

Cf. A000203, A001157, A002117, A052847, A069153, A092348.

Programs

Mathematica

Table[DivisorSigma[2,n]-DivisorSigma[1,n],{n,50}] (* Harvey P. Dale, Aug 01 2020 *)

PARI

a(n) = sigma(n,2)-sigma(n,1);

PARI

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

Formula

G.f.: Sum_{n>=1} n*(n-1) * x^n/(1-x^n). - Joerg Arndt, Jan 30 2011

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k-1)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 21 2018

From Peter Bala, Jan 21 2021: (Start)

a(n) = 2*A069153(n).

G.f.: A(x) = Sum_{n >= 1} 2*x^(2*n)/(1 - x^n)^3.

A faster converging series: A(x) = 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)).

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

a(n) = Sum_{d|n} d*(d-1). - Wesley Ivan Hurt, Aug 01 2025