Consecutive Integers with Same Divisor Sum

Theorem
Let $\sigma: \Z_{>0} \to \Z_{>0}$ be the $\sigma$ function, defined on the strictly positive integers.

The equation:
 * $\map \sigma n = \map \sigma {n + 1}$

is satisfied by integers in the sequence:
 * $14, 206, 957, 1334, 1364, 1634, 2685, 2974, 4364, 14841, 18873, \ldots$