On Integers Whose Ratio of Decimal Digit Product to Digit Sum Equals Their Divisor Count

We investigate the sequence of positive integers whose base-10 decimal representations contain no zero digits and for which the quotient of the digit product and the digit sum is an integer exactly equal to the number of positive divisors. Specifically, for an integer k, let P(k) denote the product of its decimal digits and S(k) denote the sum of its decimal digits. We study the solutions to the Diophantine equation P(k) / S(k) = tau(k), where tau(k) denotes the number of positive divisors of k. We prove the unique existence of a single-digit solution, establish the complete absence of two-digit solutions via arithmetic-geometric mean bounds, demonstrate permutation asymmetry across digit sets, tabulate the first 50 verified terms of the sequence along with their complete prime factorizations, and present open conjectures regarding asymptotic density and permutation limits.