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A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself. (A divisor d of a number n is a unitary divisor if d and n/d share no common factors). The number 6 is the only number that is both a perfect number and a unitary perfect number.

Known examples

The number 60 is a unitary perfect number because 1, 3, 4, 5, 12, 15, and 20 are its proper unitary divisors, and 1 + 3 + 4 + 5 + 12 + 15 + 20 = 60. The first five, and only known, unitary perfect numbers are:

• ${\displaystyle 6=2\times 3}$
• ${\displaystyle 60=2^{2}\times 3\times 5}$
• ${\displaystyle 90=2\times 3^{2}\times 5}$
• ${\displaystyle 87360=2^{6}\times 3\times 5\times 7\times 13}$, and
• ${\displaystyle 146361946186458562560000=2^{18}\times 3\times 5^{4}\times 7\times 11\times 13\times 19\times 37\times 79\times 109\times 157\times 313}$ (sequence A002827 in the OEIS).

The respective sums of their proper unitary divisors are as follows:

• 6 = 1 + 2 + 3
• 60 = 1 + 3 + 4 + 5 + 12 + 15 + 20
• 90 = 1 + 2 + 5 + 9 + 10 + 18 + 45
• 87360 = 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 64 + 65 + 91 + 105 + 192 + 195 + 273 + 320 + 448 + 455 + 832 + 960 + 1344 + 1365 + 2240 + 2496 + 4160 + 5824 + 6720 + 12480 + 17472 + 29120
• 146361946186458562560000 = 1 + 3 + 7 + 11 + ... + 13305631471496232960000 + 20908849455208366080000 + 48787315395486187520000 (4095 divisors in the sum)

Properties

There are no odd unitary perfect numbers. This follows since 2 divides the sum of the unitary divisors of an odd number n, where d*(n) is the number of distinct prime factors of n. One gets this because the sum of all the unitary divisors is a multiplicative function and one has that the sum of the unitary divisors of a prime power p is p + 1 which is even for all odd primes p. Therefore, an odd unitary perfect number must have only one distinct prime factor, and it is not hard to show that a power of prime cannot be a unitary perfect number, since there are not enough divisors.

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 distinct odd prime factors.

References

1. Wall, Charles R. (1988). "New unitary perfect numbers have at least nine odd components". Fibonacci Quarterly. 26 (4): 312–317. doi:10.1080/00150517.1988.12429611. ISSN 0015-0517. MR 0967649. Zbl 0657.10003.

• Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer-Verlag. pp. 84–86. ISBN 0-387-20860-7. Section B3.
• Paulo Ribenboim (2000). My Numbers, My Friends: Popular Lectures on Number Theory. Springer-Verlag. p. 352. ISBN 0-387-98911-0.
• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. ISBN 1-4020-4215-9. Zbl 1151.11300.
• Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. ISBN 1-4020-2546-7. Zbl 1079.11001.

v t e Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Descartes Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable (Triple) Sociable Betrothed
Base-dependent	Equidigital Extravagant Frugal Harshad Polydivisible Smith
Other sets	Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Refactorable Superperfect

• Integer sequences
• Perfect numbers