Quintuplets of Consecutive Integers which are not Sigma Values

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Theorem

The elements of the following $5$-tuples of consecutive integers have the property that they are not values of the $\sigma$ function $\map \sigma n$ for any $n$:

$\tuple {49, 50, 51, 52, 53}$
$\tuple {115, 116, 117, 118, 119}$
$\tuple {145, 146, 147, 148, 149}$


Proof


Sources