Consecutive Integers with Same Divisor Sum
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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
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $14$