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A344221 a(n) = Sum_{k=1..n} tau(gcd(k,n)^3), where tau(n) is the number of divisors of n.

%I A344221 #19 Nov 16 2022 06:55:32
%S A344221 1,5,6,13,8,30,10,29,21,40,14,78,16,50,48,61,20,105,22,104,60,70,26,
%T A344221 174,43,80,66,130,32,240,34,125,84,100,80,273,40,110,96,232,44,300,46,
%U A344221 182,168,130,50,366,73,215,120,208,56,330,112,290,132,160,62,624,64,170,210,253,128,420
%N A344221 a(n) = Sum_{k=1..n} tau(gcd(k,n)^3), where tau(n) is the number of divisors of n.
%H A344221 Seiichi Manyama, <a href="/A344221/b344221.txt">Table of n, a(n) for n = 1..10000</a>
%F A344221 a(n) = Sum_{d|n} phi(n/d) * tau(d^3).
%F A344221 a(n) = n * Sum_{d|n} 3^omega(d) / d.
%F A344221 If p is prime, a(p) = 3 + p.
%F A344221 From _Amiram Eldar_, Nov 15 2022: (Start)
%F A344221 Multiplicative with a(p^e) = (p^e*(p + 2) - 3)/(p - 1).
%F A344221 Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 2/p^2) = 1.8019184198... . (End)
%t A344221 Table[Sum[DivisorSigma[0,GCD[k,n]^3],{k,n}],{n,100}] (* _Giorgos Kalogeropoulos_, May 13 2021 *)
%t A344221 f[p_, e_] := (p^e*(p + 2) - 3)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 15 2022 *)
%o A344221 (PARI) a(n) = sum(k=1, n, numdiv(gcd(k, n)^3));
%o A344221 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d^3));
%o A344221 (PARI) a(n) = n*sumdiv(n, d, 3^omega(d)/d);
%Y A344221 Cf. A060648, A072691, A344222, A344223, A344224, A344225.
%K A344221 nonn,mult
%O A344221 1,2
%A A344221 _Seiichi Manyama_, May 12 2021