## Divisibility comparison between factorials

### Problem 383

Published on Saturday, 5th May 2012, 11:00 pm; Solved by 309; Difficulty rating: 50%
Let f_{5}(`n`) be the largest integer `x` for which 5^{x} divides `n`.

For example, f_{5}(625000) = 7.

Let T_{5}(`n`) be the number of integers `i` which satisfy f_{5}((2·`i`-1)!) < 2·f_{5}(`i`!) and 1 ≤ `i` ≤ `n`.

It can be verified that T_{5}(10^{3}) = 68 and T_{5}(10^{9}) = 2408210.

Find T_{5}(10^{18}).