## Geometric Progression with Maximum Sum

### Problem 542

Let `S`(`k`) be the sum of three or more distinct positive integers having the following properties:

- No value exceeds
`k`. - The values form a
**geometric progression**. - The sum is maximal.

`S`(4) = 4 + 2 + 1 = 7

`S`(10) = 9 + 6 + 4 = 19

`S`(12) = 12 + 6 + 3 = 21

`S`(1000) = 1000 + 900 + 810 + 729 = 3439

Let $T(n) = \sum_{k=4}^n (-1)^k S(k)$.

`T`(1000) = 2268

Find `T`(10^{17}).