Geometric Progression with Maximum Sum

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 = 7S(10) = 9 + 6 + 4 = 19S(12) = 12 + 6 + 3 = 21S(1000) = 1000 + 900 + 810 + 729 = 3439

Let $T(n) = \sum_{k=4}^n (-1)^k S(k)$.T(1000) = 2268

Find T(10¹⁷).

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