So, I wondered if the property described in the title (namely, the property that the sum of the divisors of $n$ equals the sum of the divisors of $n+1$) ever occurred, and went to compute it. Here are the numbers with this property up to 20.000 (including):
14, 206, 957, 1334, 1364, 1634, 2685, 2974, 4364, 14841, 18873, 19358, ...
Can anyone explain this growth? Are there infinitely many of them? (sure looks like so). Is there a formula for the nth term of this sequence, or something?