What is a single-prime function other than $f(x)=x!$? [Noob warning]: I am not a mathematician. If you use jargon, please explain or reference.
Other than $f(x)=x!$, what is a univariate non-piecewise function with a domain that is either all integers, or an infinite-sized subset of all integers, and whose range contains only integers and exactly one prime number?
For those that prefer lists, here are the criteria again:


*

*Univariate (one independent variable)

*Not $f(x)=x!$ (of which I 'think' meets the criteria below...)

*Non-piecewise (non-hybrid)

*Domain is either all integers or an infinite-sized subset of all integers

*Range contains only integers

*Range contains exactly one prime number


$$f(x)=x^2+x$$
More generally, if $g(x)$ is any function with $f(1)=p$ is prime and $f(x) \geq 2$ for all $x$ then 
$$f(x)=xg(x)$$
Also
$$h(n)= lcm [1,2,3,..,n]$$
Also
$$u(n)=n^{n-1}$$
