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A360428 Inverse Mobius transformation of A338164.

%I A360428 #21 Jan 21 2024 02:15:42
%S A360428 1,7,17,40,49,119,97,208,225,343,241,680,337,679,833,1024,577,1575,
%T A360428 721,1960,1649,1687,1057,3536,1825,2359,2673,3880,1681,5831,1921,4864,
%U A360428 4097,4039,4753,9000,2737,5047,5729,10192,3361,11543,3697,9640,11025,7399,4417,17408,7105,12775
%N A360428 Inverse Mobius transformation of A338164.
%H A360428 Amiram Eldar, <a href="/A360428/b360428.txt">Table of n, a(n) for n = 1..10000</a>
%F A360428 a(n) = Sum_{d|n} A008683(n/d)*A000005(d)*d^2.
%F A360428 Dirichlet convolution of A034714 and A008683.
%F A360428 Dirichlet g.f.: zeta^2(s-2)/zeta(s).
%F A360428 From _Amiram Eldar_, Feb 09 2023: (Start)
%F A360428 Multiplicative with a(p^e) = (e + 1 - e/p^2)*p^(2*e).
%F A360428 Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - 1/3 - zeta'(3)/zeta(3)) * n^3 / (3*zeta(3)), where gamma is Euler's constant (A001620). (End)
%F A360428 From _Peter Bala_, Jan 16 2024: (Start)
%F A360428 a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n)^2. Cf. A069097.
%F A360428 a(n) = Sum_{d divides n} d^2 * J_2(n/d), where J_2(n) = A007434(n). (End)
%p A360428 A360428 := proc(n)
%p A360428     add(numtheory[mobius](n/d)*numtheory[tau](d)*d^2,d=numtheory[divisors](n)) ;
%p A360428 end proc:
%t A360428 f[p_, e_] := (e + 1 - e/p^2)*p^(2*e); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Feb 09 2023 *)
%Y A360428 Cf. A000005, A001620, A007434, A008683, A018804, A069097, A338164.
%K A360428 nonn,mult,easy
%O A360428 1,2
%A A360428 _R. J. Mathar_, Feb 07 2023