On Divisibility-Constrained Partitions of an Arbitrary Integer

Abstract

We consider integer partitions for which the sum of the parts divides their product. Five OEIS counting functions are examined: the fixed-four-part count A343320 and the unrestricted-length counts A057568, A379733, A379734, and A379735, with distinctness and exclusion of the part 1 treated separately. We derive an exact double-sum formula for four-part partitions by reducing the remaining part to quadratic congruences and give closed formulas at odd prime squares. Each unrestricted-length count admits exact identities at twice odd primes and at prime squares. Finite valuation algebras yield coefficient formulas for all positive integer inputs, and logarithmic derivatives give exact recurrences. With the factorization of the input supplied, a formal-series method gives a conditional arithmetic-operation upper bound. The proofs are independent of the finite computational comparisons supplied for verification.