= Bi-unitary divisor
A positive <divisor> $d$ of a <positive integer> $n$ is bi-unitary when the <greatest common unitary divisor> of $d$ and $n/d$ is $1$. The comparison uses <unitary divisors>, so it differs from requiring the ordinary <greatest common divisor> to be one.
For $n=p^e$ with $p$ a <prime number> and $e\ge1$, consider $d=p^a$, where $0\le a\le e$. The only possible nontrivial common <unitary divisor> of $p^a$ and $p^{e-a}$ occurs when $a=e-a>0$. Consequently every <divisor> is bi-unitary except $p^{e/2}$ when $e$ is even.
For example, the bi-unitary <divisors> of $64=2^6$ are $1,2,4,16,32,64$; only $8$ is excluded. Its <unitary divisors> are just $1$ and $64$, showing that these two notions differ substantially. For a general <positive integer>, apply the prime-power condition independently to each <prime factor> in its <prime factorization>.
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