If you get a list of prime factors in ascending order, then I would take every other member of the list, multiply them together and start with that.

In your example, that would give you 20, which isn't too far from the correct result of 31.

Hmm .. Actually, an even better answer would be

• for a list with an odd number of elements, take all of the odd position numbers and multiply them together;
• for a list with an even number of elements, take the odd numbered elements from 1 to (n/2)-1, and the even numbered elements from n down to (n/2)+2, and multiply them together with the harmonic mean of the (n-1)/2 and n/2 elements.

For your example, that would be 2 * sqrt(2*5) * 5, which turns out to be exactly the correct answer, 31.622..

To try out the odd number, we'll try out 2000, which gives us a list of (2, 2, 2, 2, 5, 5, 5) and a result of 2 * 2 * 5 * 5 or 100. Hmm, a little high.

Well, that's a fascinating question, and good luck with that.