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Geometric Progression with Maximum Sum

Problem 542 Published on Saturday, 9th January 2016, 01:00 pm; Solved by 163;
Difficulty rating: 65%

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(1017).