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A308474 a(n) = Sum_{k=1..n^2, gcd(n,k) = 1} k. 0

%I

%S 1,4,27,64,250,216,1029,1024,2187,2000,6655,3456,13182,8232,13500,

%T 16384,39304,17496,61731,32000,55566,53240,133837,55296,156250,105456,

%U 177147,131712,341446,108000,446865,262144,359370,314432,514500,279936,911754,493848,711828,512000

%N a(n) = Sum_{k=1..n^2, gcd(n,k) = 1} k.

%F G.f.: Sum_{k>=1} mu(k)*k^3*x^k*(1 + 7*x^k + 4*x^(2*k))/(1 - x^k)^5.

%F a(n) = n^3*phi(n)/2 for n > 1.

%F a(n) = n^3 * Sum_{d|n} mu(n/d)*(d + 1)/2.

%F a(n) = A000290(n)*A023896(n).

%F a(n) = A000578(n)*A023022(n) for n > 2.

%F Sum_{k=1..n} a(k) ~ 3*n^5/(5*Pi^2). - _Vaclav Kotesovec_, May 30 2019

%t a[n_] := Sum[If[GCD[n, k] == 1, k, 0], {k, 1, n^2}]; Table[a[n], {n, 1, 40}]

%t nmax = 40; CoefficientList[Series[Sum[MoebiusMu[k] k^3 x^k (1 + 7 x^k + 4 x^(2 k))/(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t Join[{1}, Table[n^3 EulerPhi[n]/2, {n, 2, 40}]]

%t Table[n^3 Sum[MoebiusMu[n/d] (d + 1)/2, {d, Divisors[n]}], {n, 1, 40}]

%o (PARI) a(n) = sum(k=1, n^2, if (gcd(n,k)==1, k)); \\ _Michel Marcus_, May 31 2019

%Y Cf. A000010, A000290, A000578, A002618, A008683, A023022, A023896.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, May 29 2019

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Last modified October 21 08:47 EDT 2019. Contains 328292 sequences. (Running on oeis4.)