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Unitary divisor

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

Descartes number

A Descartes number is a number which is close to being a perfect number. They are named for René Descartes who observed that the number

$$D= 198585576189 = 3^2⋅7^2⋅11^2⋅13^2⋅22021 $$

would be an odd perfect number if only 22021 were a prime number, since the sum-of-divisors function for $D$ satisfies

$$\sigma(D) = (3^2+3+1)\cdot(7^2+7+1)\cdot(11^2+11+1)\cdot(13^3+13+1)\cdot(22021+1) \ . $$

A Descartes number is defined as an odd number $n = m p$ where $m$ and $p$ are coprime and $$2n = \sigma(m)\cdot(p+1)$. The example given is the only one currently known. If $m$ is an odd [[almost perfect number]], that is, $\sigma(m) = 2m-1$, then $m(2m−1)$ is a Descartes number.

References

• Banks, William D.; Güloğlu, Ahmet M.; Nevans, C. Wesley; Saidak, Filip (2008). "Descartes numbers". In De Koninck, Jean-Marie; Granville, Andrew; Luca, Florian (edd). Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006. CRM Proceedings and Lecture Notes 46 (Providence, RI: American Mathematical Society) pp. 167–173. ISBN 978-0-8218-4406-9. Zbl 1186.11004.