# Definition:Aliquot Sum

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## Definition

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

The **aliquot sum** of $n$ is defined as the sum of its aliquot parts.

### Sequence of Aliquot Sums

The sequence of aliquot sums begins:

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

## 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 part** is considered archaic nowadays, it has the advantage of being short and euphonious.

## Also see

## Linguistic Note

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

## Sources

- Weisstein, Eric W. "Restricted Divisor Function." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/RestrictedDivisorFunction.html