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- Q2370031 subject Q5519261.
- Q2370031 abstract "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.Thus, 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 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)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. 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.".
- Q2370031 wikiPageWikiLink Q1048447.
- Q2370031 wikiPageWikiLink Q12503.
- Q2370031 wikiPageWikiLink Q170043.
- Q2370031 wikiPageWikiLink Q1704865.
- Q2370031 wikiPageWikiLink Q176916.
- Q2370031 wikiPageWikiLink Q23488.
- Q2370031 wikiPageWikiLink Q239346.
- Q2370031 wikiPageWikiLink Q49008.
- Q2370031 wikiPageWikiLink Q50708.
- Q2370031 wikiPageWikiLink Q5519261.
- Q2370031 wikiPageWikiLink Q79998.
- Q2370031 comment "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.Thus, 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.".
- Q2370031 label "Unitary perfect number".