%I #25 Jun 24 2024 08:47:57

%S 1,5,12,24,35,63,70,112,135,185,176,312,247,371,450,512,425,729,532,

%T 920,903,935,782,1488,1125,1313,1458,1848,1247,2475,1426,2304,2277,

%U 2261,2660,3672,2035,2831,3198,4400,2501,4977,2752,4664,5265,4163,3290,6912,4459,6125,5508,6552

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

%H G. C. Greubel, <a href="/A344510/b344510.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = n * (n + A018804(n))/2.

%F a(n) = (n/2) * (n + Sum_{d|n} phi(n/d) * d).

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

%t a[n_] := Sum[k * GCD[k, n], {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, May 21 2021 *)

%t A344510[n_]:= (n/2)*DivisorSum[n, (#+1)*EulerPhi[n/#] &];

%t Table[A344510[n], {n,60}] (* _G. C. Greubel_, Jun 24 2024 *)

%o (PARI) a(n) = sum(k=1, n, k*gcd(k, n));

%o (PARI) a(n) = n*sumdiv(n, d, eulerphi(n/d)*(d+1))/2;

%o (Magma)

%o A344510:= func< n | (n/2)*(&+[(d+1)*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;

%o [A344510(n): n in [1..60]]; // _G. C. Greubel_, Jun 24 2024

%o (SageMath)

%o def A344510(n): return (n/2)*sum((k+1)*euler_phi(int(n//k)) for k in (1..n) if (k).divides(n))

%o [A344510(n) for n in range(1,61)] # _G. C. Greubel_, Jun 24 2024

%Y Cf. A000217, A018804, A209295, A344508.

%K nonn

%O 1,2

%A _Seiichi Manyama_, May 21 2021