cp's OEIS Frontend

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)

a(n) = A000203(n) * A001157(n).

Multiplicative with a(p^e) = (p^(e+1)-1) * (p^(2*e+2)-1) / ((p-1) * (p^2-1)).

Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-2) * zeta(s-3) / zeta(2*s-3).

In general, Dirichlet g.f. of sigma_i(n) * sigma_j(n): zeta(s) * zeta(s-i) * zeta(s-j) * zeta(s-i-j) / zeta(2*s-i-j) (Ramanujan, 1916).

Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(2) * zeta(3) * zeta(4) / zeta(5) = zeta(3) * Pi^6 / (540*zeta(5)) = 2.06386841111121962734... .

In general, Sum_{k=1..n} sigma_i(k) * sigma_j(k) ~ c(i,j) * n^(i+j+1) / (i+j+1), for i, j >= 1, where c(i,j) = zeta(i+1) * zeta(j+1) * zeta(i+j+1) / zeta(i+j+2).

G.f.: Sum_{k>=1} Sum_{l>=1} k*l^2*x^lcm(k, l)/(1 - x^lcm(k, l)). - Miles Wilson, Jul 10 2025