Proper-divisor sum
= Proper-divisor sum
{title2=$s(n)=\sigma(n)-n$}
= Aliquot sum
{synonym}
= Sum of proper divisors
{synonym}
The <proper-divisor sum>, or <aliquot sum>, adds the positive <proper divisors>:
$$
s(n)=\sum_{\substack{d\mid n\\d<n}}d=\sigma(n)-n.
$$
In particular $s(1)=0$, an <empty sum>. Removing $n$ from the <sum-of-divisors function> is essential: $\sigma(6)=12$, but its <proper-divisor sum> is $6$.