2 and 9 are the only biunitary superperfect numbers

What is it about?

A divisor d of an integer $N$ is called a unitary divisor of $N$ if $d$ and $N/d$ are relatively prime and a biunitary divisor of $N$ if $d$ and $N/d$ has no common unitary divisor except the unity. We write the sum of biunitary divisors of $N$ for $\sigma^{**}(N)$. An integer $N$ is called a biunitary superperfect number if $\sigma^{**}(\sigma^{**}(N)) = 2N$. We show that 2 and 9 are the only biunitary superperfect numbers.

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Why is it important?

There are a plenty of unsolved problems concerning numbers with special divisor-related properties such as perfect numbers, numbers $N$ such that the sum $\sigma(N)$ of the divisors of $N$ is equal to $2N$. Typical instances are determining all numbers with a given properties (or showing that there are no such numbers). It is unknown whether there are any odd perfect numbers, odd superperfect numbers (numbers satisfying $\sigma(\sigma(N))=2N$). In contrast, it is known that $6, 60, 90$ are the only biunitary perfect numbers, which satisfy $\sigma^{**}(N)=2N$. Now we have a new divisor-related special property all integers with which are determined.

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