Problem Archive

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 $T(10^{17})$.