Counting restricted colored base-3 partitions of (3^n - 3)/2 yields exactly the Euclid-Euler form 2^(n-1)(2^n - 1), so every even perfect number appears as such a partition count.
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Colored base-3 partitions, sequences of polynomials, and perfect numbers
Counting restricted colored base-3 partitions of (3^n - 3)/2 yields exactly the Euclid-Euler form 2^(n-1)(2^n - 1), so every even perfect number appears as such a partition count.