A319086 - OEIS

%I #27 Oct 21 2023 17:21:52

%S 1,13,49,161,311,743,1135,2095,3148,4948,6400,10432,12798,17502,22902,

%T 30838,36040,48676,55896,72696,86808,104232,116928,151488,170863,

%U 199255,228415,272319,297549,362349,393101,457613,509885,572309,631109,749045,801067

%N a(n) = Sum_{k=1..n} k^2*sigma(k), where sigma is A000203.

%C In general, for m>=1, Sum_{k=1..n} k^m * sigma(k) = Sum_{k=1..n} k^(m+1) * (Bernoulli(m+1, floor(1 + n/k)) - Bernoulli(m+1, 0)) / (m+1), where Bernoulli(n,x) are the Bernoulli polynomials. - _Daniel Suteu_, Nov 08 2018

%H Seiichi Manyama, <a href="/A319086/b319086.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) ~ Pi^2 * n^4/24.

%F a(n) = Sum_{k=1..n} ((k/2) * floor(n/k) * floor(1 + n/k))^2. - _Daniel Suteu_, Nov 07 2018

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

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

%o (Python)

%o def A319086(n): return sum((k*(m:=n//k)*(m+1)>>1)**2 for k in range(1,n+1)) # _Chai Wah Wu_, Oct 20 2023

%o (Python)

%o from math import isqrt

%o def A319086(n): return ((-((s:=isqrt(n))*(s+1))**3*(2*s+1)>>1) + sum(k**2*(q:=n//k)*(q+1)*(2*k*(2*q+1)+3*q*(q+1)) for k in range(1,s+1)))//12 # _Chai Wah Wu_, Oct 21 2023

%Y Cf. A000203, A024916, A143128.

%K nonn

%O 1,2

%A _Vaclav Kotesovec_, Sep 10 2018