OFFSET
1,2
COMMENTS
Inverse Mobius transform of A060724. - R. J. Mathar, Oct 15 2011
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..5000
FORMULA
a(n) = Sum_{i|n, j|n} sigma(i)*sigma(j)/sigma(lcm(i,j)), where sigma(n) = sum of divisors of n.
a(n) = Sum_{i|d, j|d} sigma(gcd(i, j));
a(n) = Sum_{d|n} d*tau(n/d)^2, where tau(n) = number of divisors of n.
Multiplicative with a(p^e) = (1-p^(3+e)-p^(2+e)+e^2+4*p^2+p^2*e^2+2*e-3*p+4*p^2*e-6*e*p-2*e^2*p)/(1-p)^3.
Dirichlet g.f.: (zeta(s))^4*zeta(s-1)/zeta(2*s). - R. J. Mathar, Feb 09 2011
G.f.: Sum_{k>=1} tau(k)^2*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Nov 02 2018
Sum_{k=1..n} a(k) ~ 5 * Pi^4 * n^2 / 144. - Vaclav Kotesovec, Jan 28 2019
a(n) = Sum_{d|n} tau(d^2)*sigma(n/d), where tau(n) = number of divisors of n, and sigma(n) = sum of divisors of n. - Ridouane Oudra, Aug 25 2019
MATHEMATICA
a[n_] := Sum[ DivisorSigma[1, i]*DivisorSigma[1, j] / DivisorSigma[1, LCM[i, j]], {i, Divisors[n]}, {j, Divisors[n]}]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 26 2013 *)
PROG
(PARI) a(n) = sumdiv(n, d, d*numdiv(n/d)^2); \\ Michel Marcus, Nov 03 2018
(Magma) [&+[d*#Divisors(Floor(n/d))^2:d in Divisors(n)]:n in [1..60]]; // Marius A. Burtea, Aug 25 2019
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Vladeta Jovovic, Jul 07 2001
STATUS
approved