Integers Differing by 2 with Same Divisor Sum

Theorem

Let $\sigma_1: \Z_{>0} \to \Z_{>0}$ be the divisor sum function, defined on the strictly positive integers.

The equation:

$\map {\sigma_1} n = \map {\sigma_1} {n + 2}$

is satisfied by integers in the sequence:

$33, 54, 284, 366, 834, 848, 918, 1240, 1504, 2910, 2913, 3304, \ldots$

This sequence is A007373 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Examples

$\sigma_1$ of $33$ equals $\sigma_1$ of $35$

$\map {\sigma_1} {33} = \map {\sigma_1} {35} = 48$

$\sigma_1$ of $54$ equals $\sigma_1$ of $56$

$\map {\sigma_1} {54} = \map {\sigma_1} {56} = 120$

$\sigma_1$ of $284$ equals $\sigma_1$ of $286$

$\map {\sigma_1} {284} = \map {\sigma_1} {286} = 504$

$\sigma_1$ of $366$ equals $\sigma_1$ of $368$

$\map {\sigma_1} {366} = \map {\sigma_1} {368} = 744$