Weird Number/Examples/70

Example of a Weird Number

$70$ is a weird number:

The aliquot sum of $70$ is $74$, but no subset of its aliquot parts add to $70$.

Proof

From $\sigma_1$ of $70$:

$\map {\sigma_1} {70} = 144$

where $\sigma_1$ denotes the divisor sum.

The aliquot sum of $70$ is given by:

$\map {\sigma_1} {70} - 70 = 144 - 70 = 74$

The aliquot parts of $70$ are:

$1, 2, 5, 7, 10, 14, 35$

Note that $4$ is not an aliquot part of $70$.

Thus $70 = 74 - 4$ cannot be made by any combination of those aliquot parts.

Hence the result.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $70$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $70$