Unitary divisor

2020 Mathematics Subject Classification: Primary: 11A [MSN][ZBL]

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 functions 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 factors 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. ISSN 0008-4395. Zbl 0312.10004
• Wall, Charles R. "New unitary perfect numbers have at least nine odd components". Fibonacci Quarterly 26 no.4 (1988) ISSN 0015-0517. MR967649. Zbl 0657.10003