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
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..10000
Programs
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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
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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 *) -
PARI
a(n) = sumdiv(n, d, 2^omega(n/d) * d^2); \\ Daniel Suteu, Mar 07 2019
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
Extensions
Extended by R. J. Mathar, Apr 02 2009