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- Unitary_perfect_number 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)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.".
- Unitary_perfect_number wikiPageID "553027".
- Unitary_perfect_number wikiPageRevisionID "602381673".
- Unitary_perfect_number hasPhotoCollection Unitary_perfect_number.
- Unitary_perfect_number subject Category:Integer_sequences.
- Unitary_perfect_number type Abstraction100002137.
- Unitary_perfect_number type Arrangement107938773.
- Unitary_perfect_number type Group100031264.
- Unitary_perfect_number type IntegerSequences.
- Unitary_perfect_number type Ordering108456993.
- Unitary_perfect_number type Sequence108459252.
- Unitary_perfect_number type Series108457976.
- Unitary_perfect_number 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.".
- Unitary_perfect_number label "Nombre parfait unitaire".
- Unitary_perfect_number label "Unitary perfect number".
- Unitary_perfect_number label "元完全數".
- Unitary_perfect_number sameAs Nombre_parfait_unitaire.
- Unitary_perfect_number sameAs m.02p8cv.
- Unitary_perfect_number sameAs Q2370031.
- Unitary_perfect_number sameAs Q2370031.
- Unitary_perfect_number sameAs Unitary_perfect_number.
- Unitary_perfect_number wasDerivedFrom Unitary_perfect_number?oldid=602381673.
- Unitary_perfect_number isPrimaryTopicOf Unitary_perfect_number.