cp's OEIS Frontend

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

%I A086666 #47 Aug 14 2025 02:43:25
%S A086666 0,2,6,14,20,38,42,70,78,112,110,182,156,226,236,310,272,416,342,504,
%T A086666 468,574,506,790,620,808,780,994,812,1228,930,1302,1172,1396,1252,
%U A086666 1820,1332,1750,1644,2120,1640,2404,1806,2478,2288,2578,2162,3286,2394,3162
%N A086666 a(n) = sigma_2(n) - sigma_1(n).
%C A086666 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
%H A086666 Harvey P. Dale, <a href="/A086666/b086666.txt">Table of n, a(n) for n = 1..1000</a>
%H A086666 Joerg Arndt, <a href="http://arxiv.org/abs/1202.6525">On computing the generalized Lambert series</a>, arXiv:1202.6525v3 [math.CA], (2012).
%F A086666 G.f.: Sum_{n>=1} n*(n-1) * x^n/(1-x^n). - _Joerg Arndt_, Jan 30 2011
%F A086666 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
%F A086666 From _Peter Bala_, Jan 21 2021: (Start)
%F A086666 a(n) = 2*A069153(n).
%F A086666 G.f.: A(x) = Sum_{n >= 1} 2*x^(2*n)/(1 - x^n)^3.
%F A086666 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)
%F A086666 From _Amiram Eldar_, Jan 01 2025: (Start)
%F A086666 Dirichlet g.f.: zeta(s) * (zeta(s-2) - zeta(s-1)).
%F A086666 Sum_{k=1..n} a(k) ~ (zeta(3)/3) * n^3. (End)
%F A086666 a(n) = Sum_{d|n} d*(d-1). - _Wesley Ivan Hurt_, Aug 01 2025
%t A086666 Table[DivisorSigma[2,n]-DivisorSigma[1,n],{n,50}] (* _Harvey P. Dale_, Aug 01 2020 *)
%o A086666 (PARI) a(n) = sigma(n,2)-sigma(n,1);
%o A086666 (PARI) a(n) = my(f = factor(n)); sigma(f, 2) - sigma(f); \\ _Amiram Eldar_, Jan 01 2025
%Y A086666 Cf. A000203, A001157, A002117, A052847, A069153, A092348.
%K A086666 nonn,easy
%O A086666 1,2
%A A086666 _Jon Perry_, Jul 27 2003