Given the following facts:
Fact 1. For a natural number $n$, the integer logarithm of that number, denoted by $\sigma (n)$, is the sum, with repetitions, of the prime factors of $n$.
For example: $\sigma (0) = \sigma (1) = 0,\sigma (63) = 13$,...
Fact 2. $\forall n \in \mathbb{N} (n > 1 \Rightarrow n \ge \sigma (n))$
Let $\tau $ be the relation on $\mathbb{N}$ such that:
$\forall n_{1},n_{2}\in \mathbb{N} ({n_1} \tau {n_2} \Leftrightarrow {n_2} = \sigma ({n_1}))$
Let $G$ be the directed graph representing the relation $\tau$ on $\mathbb{N}$. Let $G'$ be the underlying undirected graph of $G$. Prove that there is no cycle of at least length 3 in $G'$.