Sum of Unitary Divisors of Power of Prime

Theorem

Let $n = p^k$ be the power of a prime number $p$.

Then the sum of all positive unitary divisors of $n$ is $1 + n$.

Proof

Let $d \divides n$.

By Divisors of Power of Prime, $d = p^a$ for some positive integer $a \le k$.

We have $\dfrac n d = p^{k - a}$.

Suppose $d$ is a unitary divisor of $n$.

Then $d$ and $\dfrac n d$ are coprime.

If both $a, k - a \ne 0$, $p^a$ and $p^{k - a}$ have a common divisor: $p$.

Hence either $a = 0$ or $k - a = 0$.

This leads to $d = 1$ or $p^k$.

Hence the sum of all positive unitary divisors of $n$ is:

$1 + p^k = 1 + n$

$\blacksquare$