Problem S5: Factor Solitaire

Problem DescriptionCopy Anchor Link

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 — done, total cost 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 SpecificationCopy Anchor Link

The input consists of a single integer $N \ge 1$. In at least half of the cases $N \le 50000$, in at least another quarter of the cases $N \le 500000$, and in the remaining cases $N \le 5000000$.

Output SpecificationCopy Anchor Link

Compute the minimum cost that gets you to $N$.

Sample Input 1Copy Anchor Link

15

Output for Sample Input 1Copy Anchor Link

9

Sample Input 2Copy Anchor Link

2013

Output for Sample Input 2Copy Anchor Link

91

Explanation of Output for Sample Input 2Copy Anchor Link

For example, start with 1, then get to 2, 4, 5, 10, 15, 30, 60, 61, 122, 244, 305, 610, 671, 1342, and then 2013.