There are no Odd Unitary Perfect Numbers

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Theorem

No unitary perfect numbers exist which are odd.


Proof

Let $n$ be an odd number with prime decomposition $\displaystyle n = p_1^{a_1} \cdots p_k^{a_k}$.

By Sum of Unitary Divisors of Integer, the sum of its unitary divisors is $\displaystyle \prod_{1 \mathop \le i \mathop \le k} \paren {1 + p_i^{a_i} }$.

To be a unitary perfect number, this must be equal to $2 n$.


Since each $p$ is odd, each $1 + p_i^{a_i}$ is even.

Hence $\displaystyle \prod_{1 \mathop \le i \mathop \le k} \paren {1 + p_i^{a_i} }$ is divisible by $2^k$.

Since $n$ is odd, $2 n$ is divisible by $2$ but not $4$.

Thus $k = 1$.

So $n$ is a prime power.


By Sum of Unitary Divisors of Power of Prime, the sum of its unitary divisors is $1 + n$.

This cannot be equal to $2 n$, since $n > 1$.

Therefore there are no unitary perfect numbers which are odd.

$\blacksquare$


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