Consecutive Integers with Same Sigma

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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:

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

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


Examples

$\sigma$ of $14$ equals $\sigma$ of $15$

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


$\sigma$ of $206$ equals $\sigma$ of $207$

$\sigma \left({206}\right) = \sigma \left({207}\right) = 312$


Sources