$latex \displaystyle n=p_1^{\alpha_1}p_2^{\alpha_2}...p_{\omega(n)}^{\alpha_{\omega(n)}} = \prod_{i=1}^{\omega(n)} p_i^{\alpha_i}$

Where $latex \omega(n)$ is the number of distinct prime factors of n.

The arithmetical functions can be evaluated once the factorization of $latex n$ is known, (although there are many of them that can be calculated without factorization)

In fact, the only way to

*"express some arithmetical property of*$latex n$" is that the function, must be dependant on the primes, $latex p_i$ and (or) on the coefficients, $latex \alpha_i$

So the arithmetical functions can be classified, in a psychedelic and unorthodox way of course, in:

1) Functions that depend

**only on coefficients.**

2) Functions that depend

**only on primes.**

3) Functions that depend both on

**primes and coefficients**.

This classification, mathematically speaking, seems to be useless, but it is only an alternative to the alphabetical order, when it comes to deal with this topic.

**References:**

[1]-D. Joyner, R. Kreminski, J. Turisco @ Applied Abstract Algebra The Fundamental Theorem of Arithmetic

[2]-Arithmetic Function @ Wikipedia Arithmetic Function

[3]-Prime Factor @ Wikipedia Prime Factor