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A319087 a(n) = Sum_{k=1..n} k^2*phi(k), where phi is the Euler totient function A000010.

%I A319087 #18 Jan 22 2022 23:38:37
%S A319087 1,5,23,55,155,227,521,777,1263,1663,2873,3449,5477,6653,8453,10501,
%T A319087 15125,17069,23567,26767,32059,36899,48537,53145,65645,73757,86879,
%U A319087 96287,119835,127035,155865,172249,194029,212525,241925,257477,306761,332753,369257
%N A319087 a(n) = Sum_{k=1..n} k^2*phi(k), where phi is the Euler totient function A000010.
%C A319087 Comment from _N. J. A. Sloane_, Mar 22 2020: (Start)
%C A319087 Theorem: Sum_{ 1<=i<=n, 1<=j<=n, gcd(i,j)=1 } i*j = a(n).
%C A319087 Proof: From the Apostol reference we know that:
%C A319087 Sum_{ 1<=i<=n, gcd(i,n)=1 } i = n*phi(n)/2 (see A023896).
%C A319087 We use induction on n. The result is true for n=1.
%C A319087 Then a(n) - a(n-1) = 2*Sum_{ i=1..n-1, gcd(i,n)=1 } n*i = n^2*phi(n). QED (End)
%D A319087 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, problem 16, the function phi_1(n).
%H A319087 Seiichi Manyama, <a href="/A319087/b319087.txt">Table of n, a(n) for n = 1..10000</a>
%F A319087 a(n) ~ 3*n^4 / (2*Pi^2).
%t A319087 Accumulate[Table[k^2*EulerPhi[k], {k, 1, 50}]]
%o A319087 (PARI) a(n) = sum(k=1, n, k^2*eulerphi(k)); \\ _Michel Marcus_, Sep 12 2018
%Y A319087 Cf. A000010, A002088, A011755, A023896, A053191.
%K A319087 nonn
%O A319087 1,2
%A A319087 _Vaclav Kotesovec_, Sep 10 2018