Definition:Aliquot Sum

Definition

Let $n \in \Z$ be a positive integer.

The aliquot sum $\map s n$ of $n$ is defined as the sum of its aliquot parts:

$\map s n = \map {\sigma_1} n - n$

where $\map {\sigma_1} n$ denotes the divisor sum of $n$.

Sequence of Aliquot Sums

The sequence of aliquot sums begins:

$\begin{array} {r|r}

n & \map s n \\ \hline 1 & 0 \\ 2 & 1 \\ 3 & 1 \\ 4 & 3 \\ 5 & 1 \\ 6 & 6 \\ 7 & 1 \\ 8 & 7 \\ 9 & 4 \\ 10 & 8 \\ 11 & 1 \\ 12 & 16 \\ 13 & 1 \\ 14 & 10 \\ 15 & 9 \\ 16 & 15 \end{array}$

Also known as

The aliquot sum of a positive integer is also known as the more unwieldy and hence uglier term restricted divisor function.

While the term aliquot sum is considered archaic nowadays, it has the advantage of being short and euphonious.

Also see

• Definition:Untouchable Number

• Definition:Aliquot Part

• Results about aliquot sums can be found here.

Linguistic Note

The word aliquot is a Latin word meaning a few, some, or not many.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): sigma function or $\sigma$ function: 1.
• Weisstein, Eric W. "Restricted Divisor Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RestrictedDivisorFunction.html