Can someone relate the constants in the density function of numbers whose digit sum divides its digit product to properties of these numbers?

Can someone relate the constants in the density function, of numbers whose digit sum divides its digit product to properties of these numbers?

In attempting to solve Bernardo Recaman Santos' question on the Puzzling Stack Exchange, about triplets and quadruplets (in a row) of these numbers, as an exercise for my puzzle club, I came up with a closest fit of the $\psi$ (product/sum=integer) number density (# of $\psi$ numbers per digit length L) of $$0.328e^{-0.12L} \ .$$

I got this after using PARI to figure out the count and density up to $10^{40}$.

Other similar types of numbers have density formulas similar to this, and some of them have been proved (primes and 7-smooth), and some have not.

I am wondering if anyone out there can relate the constants in this density formula, to simple functions or fractions that arise from the properties of $\psi$ numbers?

It should have something to do with combinatorics of division of digit products by digit sums and 7-smooth numbers.