Consecutive Integers 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:

$\map {\sigma_1} n = \map {\sigma_1} {n + 1}$

is satisfied by integers in the sequence:

$14, 206, 957, 1334, 1364, 1634, 2685, 2974, 4364, 14841, 18873, \ldots$

This sequence is A002961 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Examples

$\sigma_1$ of $14$ equals $\sigma_1$ of $15$

$\map {\sigma_1} {14} = \map {\sigma_1} {15} = 24$

$\sigma_1$ of $206$ equals $\sigma_1$ of $207$

$\map {\sigma_1} {206} = \map {\sigma_1} {207} = 312$

Sources

• 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.)

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $14$