Pillai showed in 1929 that the function $A(n)$ the number solutions of the equation $\phi(x)=n$ is unbounded in this paper (S. Pillai  , On some functions connected with φ(n) , Bull. Amer. Math. Soc. 35 (1929), 832–836), I'm interested to know a few about Bound of solutions of this equation $\phi(x)=n!$ which it is assigned this sequence in OEIS  , In A055506 it were claimed that if $\phi(x) = n!$, then $x$ must be a product of primes $p$ such that $p - 1$ divides $n!$ but this still unclear to me if this gives to me any validity to prove that there are finitely many solutions of the equation $\phi(x) = n!$ or not .Now I want to know if $\phi(x)=n!$ have finitely many solutions or not ? probably the same meaning of this question is to ask  :Does the number solutions of $\phi(x)=n!$ bounded  ?if yes what is its bound ?

Related question: (https://math.stackexchange.com/q/3747571/156150)

Pillai showed in 1929 that the function $A(n)$ giving the number solutions of the equation $\phi(x)=n$ is unbounded in (S. Pillai, On some functions connected with $\varphi(n)$, Bull. Amer. Math. Soc. 35 (1929), 832–836). I'm interested to know about bounds on solutions of $\phi(x)=n!$ which is assigned A055506 in OEIS, where it is claimed that if $\phi(x) = n!$, then $x$ must be a product of primes $p$ such that $p - 1$ divides $n!$. It is unclear to me if this allows me to prove that there are finitely many solutions of the equation $\phi(x) = n!$. Probably an equivalent question is to ask: is the number of solutions of $\phi(x)=n!$ bounded? If yes, what is its bound ?

Related question: https://math.stackexchange.com/q/3747571/156150

Pillai showed in 1929 that the function $A(n)$ the number solutions of the equation $\phi(n)=n$ is unbounded in this paper (S. Pillai , On some functions connected with φ(n) , Bull. Amer. Math. Soc. 35 (1929), 832–836), I'm interested to know a few about Bound of solutions of this equation $\phi(a)=n!$ which it is assigned this sequence in OEIS , In A055506 it were claimed that if $\phi(x) = n!$, then $x$ must be a product of primes $p$ such that $p - 1$ divides $n!$ but this still unclear to me if this gives to me any validity to prove that there are finitely many solutions of the equation $\phi(x) = n!$ or not .Now I want to know if $\phi(a)=n!$ have finitely many solutions or not ? probably the same meaning of this question is to ask :Does the number solutions of $\phi(a)=n!$ bounded ?if yes what is its bound ?

Related question: (https://math.stackexchange.com/q/3747571/156150)

Pillai showed in 1929 that the function $A(n)$ the number solutions of the equation $\phi(x)=n$ is unbounded in this paper (S. Pillai , On some functions connected with φ(n) , Bull. Amer. Math. Soc. 35 (1929), 832–836), I'm interested to know a few about Bound of solutions of this equation $\phi(x)=n!$ which it is assigned this sequence in OEIS , In A055506 it were claimed that if $\phi(x) = n!$, then $x$ must be a product of primes $p$ such that $p - 1$ divides $n!$ but this still unclear to me if this gives to me any validity to prove that there are finitely many solutions of the equation $\phi(x) = n!$ or not .Now I want to know if $\phi(x)=n!$ have finitely many solutions or not ? probably the same meaning of this question is to ask :Does the number solutions of $\phi(x)=n!$ bounded ?if yes what is its bound ?

Related question: (https://math.stackexchange.com/q/3747571/156150)