View source for Unitary divisor

{{MSC|11A}}

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A divisor $d$ of a natural number $n$ such that $d$ and $n/d$ are [[coprime numbers]], having no common factor other than 1.  Equivalently, $d$ is a unitary divisor of $n$ if and only if every prime factor of $d$ appears to the same power in $d$ as in $n$.

The sum of unitary divisors function is denoted by $\sigma^*(n)$. The sum of the $k$-th powers of the unitary divisors is denoted by $\sigma_k^*(n)$.  These functions  are [[multiplicative arithmetic function]]s of $n$ that are not totally multiplicative. The [[Dirichlet series]] [[generating function]] is

$$ \sum_{n\ge 1}\sigma_k^*(n) n^{-s} = \frac{\zeta(s)\zeta(s-k)}{\zeta(2s-k)} . $$

The number of unitary divisors of $n$ is $\sigma_0(n) = 2^{\omega(n)}$, where $\omega(n)$ is the number of distinct [[prime factor]]s of $n$. 

A '''unitary''' or '''unitarily perfect number''' is equal to the sum of its aliquot unitary divisors:equivalently, it is $n$ such that $\sigma^*(n) = 2n$.  A unitary perfect number must be even.  It is not known whether or not there are infinitely many unitary perfect numbers, or indeed whether there are any further examples beyond the five already known.  A sixth such number would have at least nine odd prime factors.  The five known are

$$ 6 = 2\cdot3,\ 60 = 2^2\cdot3\cdot5,\ 90 = 2\cdot3^3\cdot5,\ 87360 = 2^6\cdot3\cdot5\cdot7\cdot13, $$
and
$$ 146361946186458562560000  = 2^{18}\cdot3\cdot5^4\cdot7\cdot11\cdot13\cdot19\cdot37\cdot79\cdot109\cdot157\cdot313\ . $$

==References==
* Guy, Richard K. ''Unsolved Problems in Number Theory'', Problem Books in Mathematics, 3rd ed. (Springer-Verlag, 2004) p.84, section B3. {{ISBN|0-387-20860-7}} {{ZBL|1058.11001}} 
* Sándor, Jozsef; Crstici, Borislav (2004). ''Handbook of number theory II''. (Dordrecht: Kluwer Academic, 2004) pp. 179–327. {{ISBN|1-4020-2546-7}} {{ZBL|1079.11001}}
* Wall, Charles R. "The fifth unitary perfect number", ''Can. Math. Bull.'' '''18''' (1975) 115-122. {{ZBL|0312.10004}}
* Wall, Charles R. "New unitary perfect numbers have at least nine odd components". Fibonacci Quarterly '''26''' no.4 (1988) {{MR|967649}}. {{ZBL|0657.10003}}