Unitary perfect number

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.) Some perfect numbers are not unitary perfect numbers, and some unitary perfect numbers are not regular perfect numbers.

Examples

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:

6, 60, 90, 87360, 146361946186458562560000 (sequence A002827 in the OEIS)

The respective sums of proper unitary divisors:

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 one has 2^d*(n) dividing the sum of the unitary divisors of an odd number (where d*(n) is the number of distinct prime divisors of n). One gets this because the sum of all the unitary divisors is a multiplicative function and one has the sum of the unitary divisors of a power of a prime p^a is p^a + 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.

Unsolved problem in mathematics:

Are there infinitely many unitary perfect numbers?
(more unsolved problems in mathematics)

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.^[1]

References

↑ Wall, Charles R. (1988). "New unitary perfect numbers have at least nine odd components". Fibonacci Quarterly. 26 (4): 312–317. ISSN 0015-0517. MR 967649. 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.

Divisibility-based sets of integers

Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number

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 Erdős–Nicolas

With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird

Aliquot sequence-related	Untouchable Amicable Sociable Betrothed

Other sets	Deficient Friendly Solitary Sublime Harmonic divisor Frugal Equidigital Extravagant

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