I am asking this question out of curiosity, there is not really a particular reason behind this question.
$\Pi_\Sigma$:-
I created this function which I call the Factor-Sum function. I don't know if this function is a popular function used by mathematicians or not, so I am just calling it my own.
So I defined,
$\Pi_\Sigma(N)= \begin{cases}\sum_{i=1}^{n}{a_i}{p_i}, & \text{if}N=\prod_{i=1}^{n}{p_i}^{a_i} \\N, & \text{if} N=0,1 \\-\Pi_\Sigma(|N|), & \text{if} N<0 \end{cases}$
So this is an odd function which gives the input back when given a prime or $0$ or $1$ and gives the sum of the prime factors when given a composite.
For example:-
●$\Pi_\Sigma(13)=13$
●$\Pi_\Sigma(18)=2+3+3=8$
●$\Pi_\Sigma(-18)=-8$
Interpolation of $\Pi_\Sigma$:-
So $\operatorname{Dom}(\Pi_\Sigma)=\mathbb{Z}$
I want to extend the domain of $\Pi_\Sigma$ to $\mathbb{R}$.
I have estimated few non-integer values of the function using newton forward formula-
$f(a+hu)=f(a)+\sum_{n=1}^{\infty}\frac{u!}{n!(u-n)!}\Delta^nf(a)$
But it doesn't give an accurate or an estimated general formula for $\Pi_\Sigma$ for any real $N$.
I don't want interpolations like Peicewise constant interpolation or Linear interpolation which gives sharp edges in the graph of the function.
I want an interpolation of $\Pi_\Sigma$ such that it becomes differentiable everywhere.(Such as polynomial interpolation, spline interpolation)
Any help would be appreciated.
(I don't really know much about interpolation that's why I need help.)