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A342535 a(n) = Sum_{k=1..n} phi(gcd(k, n))^3.

%I A342535 #27 Nov 15 2022 09:16:55
%S A342535 1,2,10,11,68,20,222,78,238,136,1010,110,1740,444,680,604,4112,476,
%T A342535 5850,748,2220,2020,10670,780,8276,3480,6330,2442,21980,1360,27030,
%U A342535 4792,10100,8224,15096,2618,46692,11700,17400,5304,64040,4440,74130,11110,16184,21340,97382,6040
%N A342535 a(n) = Sum_{k=1..n} phi(gcd(k, n))^3.
%H A342535 Seiichi Manyama, <a href="/A342535/b342535.txt">Table of n, a(n) for n = 1..10000</a>
%F A342535 a(n) = Sum_{d|n} phi(n/d) * phi(d)^3.
%F A342535 a(n) = Sum_{k=1..n} phi(gcd(k,n))*phi(n/gcd(k,n))^2. - _Richard L. Ollerton_, May 10 2021
%F A342535 Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/4) * Product_{p prime} (1 - 3/p^2 + 3/p^3 - 2/p^4 + 3/p^6 - 3/p^7 + 1/p^8) = 0.093622450005... . - _Amiram Eldar_, Nov 15 2022
%t A342535 a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^3 &]; Array[a, 50] (* _Amiram Eldar_, Mar 15 2021 *)
%t A342535 Table[Sum[EulerPhi[GCD[k,n]]^3,{k,n}],{n,50}] (* _Harvey P. Dale_, Jul 15 2021 *)
%o A342535 (PARI) a(n) = sum(k=1, n, eulerphi(gcd(k, n))^3);
%o A342535 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*eulerphi(d)^3);
%Y A342535 Cf. A000010, A002117, A029935, A342470, A342534.
%K A342535 nonn,mult
%O A342535 1,2
%A A342535 _Seiichi Manyama_, Mar 15 2021