Term for the sum of products of a number's prime factors, excluding one factor per product.

This function is called the arithmetic derivative of a number.

Here's a recursive definition:

• Define $0' = 1' = 0$
• Define $p' = 1$ for any prime.
• If a number $m = a\cdot b$, define $m' = a'\cdot b + a \cdot b'$.

By induction you see that this equals your "sum of all but 1 prime factor" function.

This is the arithmetic derivative, OEIS A003415.

The arithmetic derivative $n'$ satisfies $(mn)'=m'n+mn'$ for all $m,n$, just like the ordinary derivative. From this we get $(abc)'=a'bc+ab'c+abc'$ and so on, and clearly these also apply in the cases where all the variables are prime numbers and $p'=1$.