## Uphill paths

### Problem 411

Published on Saturday, 19th January 2013, 10:00 pm; Solved by 325; Difficulty rating: 45%
Let `n` be a positive integer. Suppose there are stations at the coordinates (`x`, `y`) = (2^{i} mod `n`, 3^{i} mod `n`) for 0 ≤ `i` ≤ 2`n`. We will consider stations with the same coordinates as the same station.

We wish to form a path from (0, 0) to (`n`, `n`) such that the x and y coordinates never decrease.

Let S(`n`) be the maximum number of stations such a path can pass through.

For example, if `n` = 22, there are 11 distinct stations, and a valid path can pass through at most 5 stations. Therefore, S(22) = 5.
The case is illustrated below, with an example of an optimal path:

It can also be verified that S(123) = 14 and S(10000) = 48.

Find ∑ S(`k`^{5}) for 1 ≤ `k` ≤ 30.