Define $$f(m):=m\mod \varphi(m)$$ where $\ \varphi(m)\ $ denotes the totient function.
Now define $\ g(n)\ $ to be the smalles positive integer $\ m\ $ with $\ f(m)=n\ $ , if such a positive integer exists and undefined , if no such integer exist.
Is $\ g(n)\ $ defined for every positive integer $\ n\ $?
I calculated $\ g(n)\ $ upto $\ n=3\cdot 10^4\ $. The largest value that appeared was $$g(23455)=3796759$$ and $\ g(n)\ $ is actually defined upto this limit.