PEG Judge - CCC13S5 - Factor Solitare

2013 Canadian Computing Competition, Stage 1

Problem S5: Factor Solitare

In the game of Factor Solitaire, you start with the number 1, and try to change it to some given target number n by repeatedly using the following operation. In each step, if c is your current number, you split it into two positive factors a, b of your choice such that c = a * b. You then add a to your current number c to get your new current number. Doing this costs you b points. You continue doing this until your current number is n, and you try to achieve this at the cost of a minimum total number of points.

For example, here is one way to get to 15:

start with 1
change 1 to 1+1=2—cost so far is 1
change 2 to 2+1=3—cost so far is 1+2
change 3 to 3+3=6—cost so far is 1+2+1
change 6 to 6+6=12—cost so far is 1+2+1+1
change 12 to 12+3=15—cost so far is 1+2+1+1+4=9.

In fact, this is the minimum possible total cost to get 15. You want to compute the minimum total cost for other target end numbers.

Input

The input costs of a single integer N ≥ 1. In at least half of the cases N ≤ 5×10⁴, in at least another quarter of the cases N ≤ 5×10⁵, and in the remaining cases N ≤ 5×10⁶.