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 2d*(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 pa is pa + 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
  1. Wall, Charles R. (1988). "New unitary perfect numbers have at least nine odd components". Fibonacci Quarterly. 26 (4): 312–317. 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.
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