The sigma function
$$\sigma_{1}({p_i}^{\alpha_i}) = \Sigma_{i=0}^{\alpha_i}{{p_i}^{\alpha_i}}$$displaystyle\sum_{j=0}^{\alpha_i}{{p_i}^j}$$
satisfies the inequalities
$$\sigma_{1}({p_i}^{\alpha_i}) \gt (\alpha_i + 1)(\sqrt{p_i})^{\alpha_i}$$
$$\sigma_{1}({p_i}^{\alpha_i}) \gt 1 + \alpha_i(\sqrt{p_i})^{1 + \alpha_i}$$
for prime $p_i$ and $\alpha_i \ge 1$.
The "proof" uses the Arithmetic Mean-Geometric Mean Inequality.
As a particular application of this result, Sorli's Conjecture implies the OPN Conjecture.
