Factor Solitaire

문제

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.

입력

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

출력

Compute the minimum cost that gets you to N.