Factor $2021 = 43 \cdot 47.$ Since $\sigma$ is multiplicative, it suffices to force $p + p^2 + \cdots + p^{en} \equiv 0$ modulo $43$ and $47$ for every prime $p$ and every positive integer $e$

Primes $p \equiv 1 \pmod q$ force $q$ to divide $n,$ while primes that are primitive roots mod $q$ force $q - 1$ to divide $n;$ take the least common multiple

Note $2021 = 43 \cdot 47.$ If $a = \prod p_i^{e_i},$ then $\sigma(a^n) = \prod \sigma(p_i^{e_i n}),$ so it suffices (and is necessary, taking $a$ prime) that $\sigma(p^N) \equiv 1,$ i.e. $p + p^2 + \cdots + p^N \equiv 0$ $\pmod{43 \cdot 47},$ for every prime $p$ and every multiple $N$ of $n.$

Fix $q \in \{43, 47\}.$ If $q$ divides $p$ the sum is $0.$ If $p \equiv 1 \pmod q$ the sum is $\equiv N,$ so choosing such a prime (Dirichlet) forces $q$ to divide $n.$ Otherwise the sum is $p \cdot \frac{p^N - 1}{p - 1}$ with $p - 1$ invertible, so we need $p^N \equiv 1 \pmod q;$ choosing $p$ to be a primitive root mod $q$ forces $q - 1$ to divide $n.$ Conversely, if $q(q-1)$ divides $n$ then for every multiple $N$ of $n$ and every prime $p,$ the sum vanishes mod $q$ in all three cases. Hence the least $n$ is $$\begin{aligned} n &= \operatorname{lcm}(43 \cdot 42,\ 47 \cdot 46) \\ &= 2 \cdot 3 \cdot 7 \cdot 23 \cdot 43 \cdot 47. \end{aligned}$$

The sum of the prime factors is $2 + 3 + 7 + 23 + 43 + 47 = 125.$