OFFSET
1,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = (1/n^2)*Sum_{d|n} sigma_2(d)^2*moebius(n/d).
a(n) = Sum_{d|n} 2^omega(n/d) * d^2. - Daniel Suteu, Mar 07 2019
From Amiram Eldar, Dec 05 2022: (Start)
Multiplicative with a(p^e) = (p^(2*e)*(p^2+1) - 2)/(p^2-1).
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(3)^2/(3*zeta(6)) = 0.473436... . (End)
Dirichlet g.f.: zeta(s)^2*zeta(s-2)/zeta(2*s). - Amiram Eldar, Jan 06 2023
a(n) = Sum_{1 <= j, k <= n} tau(gcd(j, k, n)^2) = Sum_{d divides n} tau(d^2)* J_2(n/d), where the divisor function tau(n) = A000005(n) and the Jordan totient function J_2(n) = A007434(n). - Peter Bala, Jan 22 2024
a(n) = Sum_{d divides n} J_4(d)/J_2(d) = Sum_{1 <= i, j, k, l <= n} 1/(J_2(n/gcd(i,j,k,l,n))), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 23 2024
MAPLE
A158949 := proc(n) add(numtheory[sigma][2](d)^2*numtheory[mobius](n/d), d=numtheory[divisors](n))/n^2 ; end: seq( A158949(n), n=1..80) ; # R. J. Mathar, Apr 02 2009
MATHEMATICA
a[n_] := Sum[2^PrimeNu[n/d] d^2, {d, Divisors[n]}];
Array[a, 80] (* Jean-François Alcover, Nov 20 2020 *)
f[p_, e_] := (p^(2*e)*(p^2 + 1) - 2)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Dec 05 2022 *)
PROG
(PARI) a(n) = sumdiv(n, d, 2^omega(n/d) * d^2); \\ Daniel Suteu, Mar 07 2019
CROSSREFS
KEYWORD
easy,mult,nonn
AUTHOR
Vladeta Jovovic, Mar 31 2009
EXTENSIONS
Extended by R. J. Mathar, Apr 02 2009
STATUS
approved