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A342534 a(n) = Sum_{k=1..n} phi(gcd(k, n))^2.

%I A342534 #26 Nov 15 2022 09:17:00
%S A342534 1,2,6,7,20,12,42,26,50,40,110,42,156,84,120,100,272,100,342,140,252,
%T A342534 220,506,156,484,312,438,294,812,240,930,392,660,544,840,350,1332,684,
%U A342534 936,520,1640,504,1806,770,1000,1012,2162,600,2022,968,1632,1092,2756,876,2200,1092,2052
%N A342534 a(n) = Sum_{k=1..n} phi(gcd(k, n))^2.
%H A342534 Seiichi Manyama, <a href="/A342534/b342534.txt">Table of n, a(n) for n = 1..10000</a>
%F A342534 a(n) = Sum_{d|n} phi(n/d) * phi(d)^2.
%F A342534 a(n) = Sum_{k=1..n} phi(gcd(k,n))*phi(n/gcd(k,n)). - _Richard L. Ollerton_, May 10 2021
%F A342534 From _Amiram Eldar_, Nov 15 2022: (Start)
%F A342534 Multiplicative with a(p^e) = (p-1)*(p^(e-2) - p^(2*e-3) + p^(2*e-1)).
%F A342534 Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(2)/(3*zeta(3)) * Product_{p prime} (1 - (2*p-1)/p^3) = A306633 * A065464 / 3 = 0.195343... . (End)
%t A342534 a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^2 &]; Array[a, 50] (* _Amiram Eldar_, Mar 15 2021 *)
%o A342534 (PARI) a(n) = sum(k=1, n, eulerphi(gcd(k, n))^2);
%o A342534 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*eulerphi(d)^2);
%Y A342534 Cf. A000010, A029935, A029939, A065464, A306633, A338997, A342535.
%K A342534 nonn,mult
%O A342534 1,2
%A A342534 _Seiichi Manyama_, Mar 15 2021