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A natural number $d$ is a unitary divisor of a number $n$ if $d$ is a divisor of $n$ and $d$ and $n/d$ are coprime, 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:equivalen tly, it is n such that $\sigma^*(n) = 2n$. A unitary perfect number must be even and it is conjectured that there are only finitely many such. 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