Weird Number/Examples/836

Example of a Weird Number

$836$ is a weird number:

The aliquot sum of $836$ is $844$, but no subset of its aliquot parts add to $836$.

Proof

From $\sigma_1$ of $836$:

$\map {\sigma_1} {836} = 1680$

where $\sigma_1$ denotes the divisor sum function.

The aliquot sum of $836$ is given by:

$\map {\sigma_1} {836} - 836 = 1680 - 836 = 844$

The aliquot parts of $836$ are found in $\sigma_0$ of $836$:

$1, 2, 4, 11, 19, 22, 38, 44, 76, 209, 418$

Note that $8$ is not an aliquot part of $836$, and cannot be made from the aliquot parts of $836$.

Thus $836 = 844 - 8$ cannot be made by any combination of those aliquot parts.

Hence the result.

$\blacksquare$