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%I #14 Nov 26 2021 07:26:50
%S 1,4,6,8,10,24,14,12,15,40,22,48,26,56,60,16,34,60,38,80,84,88,46,72,
%T 35,104,24,112,58,240,62,20,132,136,140,120,74,152,156,120,82,336,86,
%U 176,150,184,94,96,63,140,204,208,106,96,220,168,228,232,118,480
%N Dirichlet convolution of the squarefree kernel function (A007947) with itself.
%F Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + p^(1-s) - p^(-s))^2.
%F a(n) = Sum_{d|n} A007947(d) * A007947(n/d).
%F a(n) = Sum_{d|n} abs(A097945(d)) * A191750(n/d).
%F Multiplicative with a(p^e) = (e-1)*p^2 + 2*p. - _Amiram Eldar_, Nov 25 2021
%F From _Vaclav Kotesovec_, Nov 26 2021: (Start)
%F Dirichlet g.f.: zeta(s-1)^2 * zeta(s)^2 * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s))^2.
%F Let f(s) = Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)), then
%F Sum_{k=1..n} a(k) ~ Pi^2 * f(2)^2 * n^2 / 144 * (Pi^2 * (2*log(n) + 4*gamma - 1 + 4*f'(2)/f(2)) + 24*zeta'(2)), where f(2) = Product_{primes p} (1 - 2/p^2 + 1/p^3) = A065464 = 0.428249505677094440218765707581823546121298513355936144..., f'(2) = f(2) * Sum_{primes p} log(p) * (3*p - 2) / (p^3 - 2*p + 1) = 0.6293283828324697510445630056425352981207558777167836747744750359407300858..., zeta'(2) = -A073002 and gamma is the Euler-Mascheroni constant A001620. (End)
%t Table[Sum[Last[Select[Divisors[d], SquareFreeQ]] Last[Select[Divisors[n/d], SquareFreeQ]], {d, Divisors[n]}], {n, 1, 60}]
%t f[p_, e_] := (e - 1)*p^2 + 2*p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* _Amiram Eldar_, Nov 25 2021 *)
%o (PARI)
%o A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
%o A349694(n) = sumdiv(n,d,A007947(n/d)*A007947(d)); \\ _Antti Karttunen_, Nov 25 2021
%Y Cf. A007947, A097945, A176345, A191750.
%K nonn,mult
%O 1,2
%A _Ilya Gutkovskiy_, Nov 25 2021
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