A319085 - OEIS

%I #29 Jan 19 2025 00:35:18

%S 1,9,27,75,125,269,367,623,866,1266,1508,2372,2710,3494,4394,5674,

%T 6252,8196,8918,11318,13082,15018,16076,20684,22559,25263,28179,32883,

%U 34565,41765,43687,49831,54187,58811,63711,75375,78113,83889,89973,102773,106135

%N a(n) = Sum_{k=1..n} k^2*tau(k), where tau is A000005.

%C In general, for m>=1, Sum_{k=1..n} k^m * tau(k) = Sum_{k=1..n} k^m * (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 Amiram Eldar, <a href="/A319085/b319085.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) ~ n^3 * (log(n) + 2*gamma - 1/3)/3, where gamma is the Euler-Mascheroni constant A001620.

%F a(n) = Sum_{k=1..n} k^2 * Bernoulli(3, floor(1 + n/k)) / 3, where Bernoulli(n,x) are the Bernoulli polynomials. - _Daniel Suteu_, Nov 08 2018

%F a(n) = Sum_{k=1..n} Sum_{i=1..floor(n/k)} i^2 * k^2. - _Wesley Ivan Hurt_, Nov 26 2020

%t Accumulate[Table[k^2*DivisorSigma[0, k], {k, 1, 50}]]

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

%o (PARI) f(n) = n*(n+1)*(2*n+1)/6; \\ A000330

%o a(n) = 2*sum(k=1, sqrtint(n), k^2 * f(n\k)) - f(sqrtint(n))^2; \\ _Daniel Suteu_, Nov 26 2020

%o (Python)

%o from math import isqrt

%o def A319085(n): return (-((s:=isqrt(n))*(s+1)*(2*s+1))**2//12 + sum(k**2*(q:=n//k)*(q+1)*(2*q+1) for k in range(1,s+1)))//3 # _Chai Wah Wu_, Oct 21 2023

%Y Cf. A000005, A006218, A034714, A143127.

%K nonn,changed

%O 1,2

%A _Vaclav Kotesovec_, Sep 10 2018