Number of sums of factors Given a positive integer n > 1 determine how many numbers can be made by adding integers greater than 1 whose product is n.  For example if n = 24 we can express n as a product in the following ways

24 = 24             -&gt; 24            = 24
24 = 12 * 2         -&gt; 12 + 2        = 14
24 = 6 * 2 * 2      -&gt; 6 + 2 + 2     = 10
24 = 6 * 4          -&gt; 6 + 4         = 10
24 = 3 * 2 * 2 * 2  -&gt; 3 + 2 + 2 + 2 = 9
24 = 3 * 4 * 2      -&gt; 3 + 4 + 2     = 9
24 = 3 * 8          -&gt; 3 + 8         = 11


We can get the following numbers this way:

24, 14, 11, 10, 9


That is a total of 5 numbers, so our result is 5.

Task

Write a program or function that takes n as input and returns the number of results that can be obtained this way.

This is a code-golf question so answers will be scored in bytes, with fewer bytes being better.

OEIS sequence

OEIS A069016
Brachylog, 8 bytes

{~×≜+}ᶜ¹


Try it online!

Explanation

{    }ᶜ¹  Count unique results of this predicate:
 ~×       Create list of numbers whose product is the input.
   ≜      Label the list, forcing it to take a concrete value.
    +     Take its sum.


I'm not entirely sure why ~× only produces lists with elements above 1, but it seems to do so, which works great in this challenge.