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a(n) = Sum_{k=1..n} k^2*phi(k), where phi is the Euler totient function A000010.
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%I #18 Jan 22 2022 23:38:37

%S 1,5,23,55,155,227,521,777,1263,1663,2873,3449,5477,6653,8453,10501,

%T 15125,17069,23567,26767,32059,36899,48537,53145,65645,73757,86879,

%U 96287,119835,127035,155865,172249,194029,212525,241925,257477,306761,332753,369257

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

%C Comment from _N. J. A. Sloane_, Mar 22 2020: (Start)

%C Theorem: Sum_{ 1<=i<=n, 1<=j<=n, gcd(i,j)=1 } i*j = a(n).

%C Proof: From the Apostol reference we know that:

%C Sum_{ 1<=i<=n, gcd(i,n)=1 } i = n*phi(n)/2 (see A023896).

%C We use induction on n. The result is true for n=1.

%C 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 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, problem 16, the function phi_1(n).

%H Seiichi Manyama, <a href="/A319087/b319087.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) ~ 3*n^4 / (2*Pi^2).

%t Accumulate[Table[k^2*EulerPhi[k], {k, 1, 50}]]

%o (PARI) a(n) = sum(k=1, n, k^2*eulerphi(k)); \\ _Michel Marcus_, Sep 12 2018

%Y Cf. A000010, A002088, A011755, A023896, A053191.

%K nonn

%O 1,2

%A _Vaclav Kotesovec_, Sep 10 2018