A364269 - OEIS

%I #32 Oct 22 2023 00:45:11

%S 1,41,311,1655,4905,15705,32855,76375,142714,272714,435096,797976,

%T 1171466,1857466,2734966,4131702,5556472,8210032,10692990,15060990,

%U 19691490,26186770,32635280,44385680,54557555,69497155,85637215,108686815,129222353,164322353

%N a(n) = Sum_{k=1..n} k^3*sigma_2(k), where sigma_2 is A001157.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Faulhaber%27s_formula">Faulhaber's formula</a>.

%F a(n) = Sum_{k=1..n} k^5 * A000537(floor(n/k)).

%F a(n) ~ (zeta(3)/6) * n^6. - _Amiram Eldar_, Oct 20 2023

%t Accumulate[Table[n^3*DivisorSigma[2, n], {n, 1, 30}]] (* _Amiram Eldar_, Oct 20 2023 *)

%o (PARI) f(n, m) = (subst(bernpol(m+1, x), x, n+1)-subst(bernpol(m+1, x), x, 0))/(m+1);

%o a(n, s=3, t=2) = sum(k=1, n, k^(s+t)*f(n\k, s));

%o (Python)

%o def A364269(n): return sum(k*(k**2*(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 A364269(n): return ((((s:=isqrt(n))*(s+1))**4*(1-s*(s+1<<1))>>2) + sum(((q:=n//k)*(q+1))**2*k**3*(3*k**2+(q*(q+1<<1)-1)) for k in range(1,s+1)))//12 # _Chai Wah Wu_, Oct 21 2023

%Y Cf. A356125, A364268.

%Y Cf. A000537, A001157.

%K nonn,easy

%O 1,2

%A _Seiichi Manyama_, Oct 20 2023