We examine and classify the solutions to certain Diophantine equations involving factorials and some well known arithmetic functions. F. Luca has showed that there are finitely many solutions to the equation:
f(n!)=a m!
where f is one of the arithmetic functions φ or σ (sum of the divisors function) and a is a rational number. We study the solutions for this equation when a is a prime power or a reciprocal of a prime power. Furthermore, we prove that if ρ is prime and k>0 , then
φ(n!)=ρ k m! and ρ k f(n!)=m!
have finitely many solutions (ρ,k,m,n) , too.