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A158949 Inverse Moebius transform of A065958.

%I A158949 #34 Jan 25 2024 08:00:18
%S A158949 1,6,11,26,27,66,51,106,101,162,123,286,171,306,297,426,291,606,363,
%T A158949 702,561,738,531,1166,677,1026,911,1326,843,1782,963,1706,1353,1746,
%U A158949 1377,2626,1371,2178,1881,2862,1683,3366,1851,3198,2727,3186,2211,4686,2501
%N A158949 Inverse Moebius transform of A065958.
%H A158949 Alois P. Heinz, <a href="/A158949/b158949.txt">Table of n, a(n) for n = 1..10000</a>
%F A158949 a(n) = (1/n^2)*Sum_{d|n} sigma_2(d)^2*moebius(n/d).
%F A158949 a(n) = Sum_{d|n} 2^omega(n/d) * d^2. - _Daniel Suteu_, Mar 07 2019
%F A158949 From _Amiram Eldar_, Dec 05 2022: (Start)
%F A158949 Multiplicative with a(p^e) = (p^(2*e)*(p^2+1) - 2)/(p^2-1).
%F A158949 Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(3)^2/(3*zeta(6)) = 0.473436... . (End)
%F A158949 Dirichlet g.f.: zeta(s)^2*zeta(s-2)/zeta(2*s). - _Amiram Eldar_, Jan 06 2023
%F A158949 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
%F A158949 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
%p A158949 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
%t A158949 a[n_] := Sum[2^PrimeNu[n/d] d^2, {d, Divisors[n]}];
%t A158949 Array[a, 80] (* _Jean-François Alcover_, Nov 20 2020 *)
%t A158949 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 *)
%o A158949 (PARI) a(n) = sumdiv(n, d, 2^omega(n/d) * d^2); \\ _Daniel Suteu_, Mar 07 2019
%Y A158949 Cf. A002117, A013664, A065958.
%K A158949 easy,mult,nonn
%O A158949 1,2
%A A158949 _Vladeta Jovovic_, Mar 31 2009
%E A158949 Extended by _R. J. Mathar_, Apr 02 2009