Weird Number/Examples/4030

Example of a Weird Number

$4030$ is a weird number:

The aliquot sum of $4030$ is $4034$, but no subset of its aliquot parts add to $4030$.

Proof

From $\sigma_1$ of $4030$:

$\map {\sigma_1} {4030} = 8064$

where $\map {\sigma_1} {4030}$ denotes the divisor sum function.

The aliquot sum of $4030$ is given by:

$\map {\sigma_1} {4030} - 4030 = 8064 - 4030 = 4034$

The aliquot parts of $4030$ are found in $\sigma_0$ of $4030$:

$1, 2, 5, 10, 13, 26, 31, 62, 65, 130, 155, 310, 403, 806, 2015$

Note that $4$ is not an aliquot part of $4030$, and cannot be made from the aliquot parts of $4030$.

Thus $4030 = 4034 - 4$ cannot be made by any combination of those aliquot parts.

Hence the result.

$\blacksquare$