I have been trying to find this summation faster, Is there any sequence that can be observed?
$$Z(N)=\sum_{i=1}^N i^2\left\lfloor \frac{N}{i^2} \right\rfloor$$
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Sign up to join this communityI have been trying to find this summation faster, Is there any sequence that can be observed?
$$Z(N)=\sum_{i=1}^N i^2\left\lfloor \frac{N}{i^2} \right\rfloor$$
This sequence is OEIS' A309125 with some interesting links like A035316 for the differences obtained with following Dirichlet generating function (and much more) : $$\zeta(s)\, \zeta(2s-2)$$ Details appear at $\,3.6\,$ (for $\,t=2$) in Richard Mathar's paper "Survey of Dirichlet Series of Multiplicative Arithmetic Functions".
Using the free pari/gp we may obtain these differences and (adding) your initial sequence :
vz=vector(50,n,1)
= [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...]
vz2=vector(#vz,n,n*(sqrtint(n)^2==n))
= [1, 0, 0, 4, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 16,...]
d= dirmul(vz,vz2)
= [1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 1, 1, 1, 21, 1, 10, 1, 5, 1, 1, 1,...]
s=0;vector(#vz,n,s+=d[n])
= [1, 2, 3, 8, 9, 10, 11, 16, 26, 27, 28, 33, 34, 35, 36, 57, 58, 68, 69,...]