%I #32 Oct 24 2023 17:37:35

%S 1,6,14,31,45,81,101,150,191,253,285,401,439,527,623,752,802,979,1035,

%T 1233,1369,1509,1577,1901,2020,2186,2362,2642,2728,3136,3228,3549,

%U 3765,3983,4215,4772,4882,5126,5382,5932,6054,6630,6758,7202,7664,7960,8100,8936

%N a(n) = Sum_{k=1..n} k * floor(n/k)^2.

%H Seiichi Manyama, <a href="/A350107/b350107.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{k=1..n} k * Sum_{d|k} (2*d - 1)/d = 2 * A143127(n) - A024916(n).

%F G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * x^k/(1 - x^k)^2.

%F a(n) = Sum_{k=1..n} 2 * k * tau(k) - sigma(k).

%t a[n_] := Sum[k * Floor[n/k]^2, {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, Dec 14 2021 *)

%t Accumulate[Table[2*k*DivisorSigma[0, k] - DivisorSigma[1, k], {k, 1, 100}]] (* _Vaclav Kotesovec_, Dec 16 2021 *)

%o (PARI) a(n) = sum(k=1, n, k*(n\k)^2);

%o (PARI) a(n) = sum(k=1, n, k*sumdiv(k, d, (2*d-1)/d));

%o (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (2*k-1)*x^k/(1-x^k)^2)/(1-x))

%o (PARI) a(n) = sum(k=1, n, 2*k*numdiv(k)-sigma(k));

%o (Python)

%o from math import isqrt

%o def A350107(n): return -(s:=isqrt(n))**3*(s+1)+sum((q:=n//k)*((k<<1)*((q<<1)+1)-q-1) for k in range(1,s+1))>>1 # _Chai Wah Wu_, Oct 24 2023

%Y Column 2 of A350106.

%Y Cf. A000005 (tau), A024916, A143127, A222548, A350123, A350128.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Dec 14 2021