2023 IOQM/Problem 19

For n ∈ N , let P(n) denote the product of the digits in n and S(n) denote the sum of the digits in n . Consider the set A = {n ∈ N : P(n) is non-zero, square free and S(n) is a proper divisor of P(n) } . Find the maximum possible number of digits of the numbers in A .