A158949 Inverse Moebius transform of A065958.

1, 6, 11, 26, 27, 66, 51, 106, 101, 162, 123, 286, 171, 306, 297, 426, 291, 606, 363, 702, 561, 738, 531, 1166, 677, 1026, 911, 1326, 843, 1782, 963, 1706, 1353, 1746, 1377, 2626, 1371, 2178, 1881, 2862, 1683, 3366, 1851, 3198, 2727, 3186, 2211, 4686, 2501

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

Cf. A002117, A013664, A065958.

Sequence in context: A219628 A093027 A109296 * A061203 A263419 A140359

Adjacent sequences: A158946 A158947 A158948 * A158950 A158951 A158952

Keyword

easy,mult,nonn

Author

Vladeta Jovovic, Mar 31 2009

Extensions

Extended by R. J. Mathar, Apr 02 2009

Status

approved