Let $p$ be a prime number. The number of monic irreducible polynomial $P\in{\mathbb F}_p[X]$, in terms of the degree $d$, begins with $${\rm irr}(1)=p,\qquad{\rm irr}(2)=\frac{p(p-1)}2,\qquad{\rm irr}(3)=\frac{p(p^2-1)}3,\qquad{\rm irr}(4)=\frac{5p^2(p^2-1)}{12}.$$ This seems to be the beginning of a nice sequence of polynomials in $p$. Does someone know the general formula?

**Motivation**. As F. Brunault pointed out to me, there is a polynomial $\Pi_{n,p}$ that vanishes over ${\bf M}_n({\mathbb F}_p)$. Just take the *lcm* of all the polynomials of degree $n$ over $F_p$.
In closed form, it is the product of all the irreducible polynomials of degree $d\le n$, to the power $[\frac{n}{d}]$ (integral part). The degree of $\Pi_{n,p}$ is
$$\delta(n)=\sum_{d\le n}d[\frac{n}{d}]{\rm irr}(d).$$
In terms of $n$, this degree is
$$\delta(1)=p,\quad\delta(2)=p(p+1),\quad\delta(3)=p(p^2+p+1),\quad\delta(4)=\frac13p(p+1)(5p^2-2p+3).$$

**Edit**. The values given above for $n=4$ are erroneous.