“Unitary perfect number”的意思、由来-开放百科全书

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Unitary perfect number

释义

Examples
Properties
References

A unitary perfect number is an integer which is the sum of it’s 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 {{OEIS|id=A002827}}

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|mathematics|Are there infinitely many unitary perfect numbers?}}

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. ^{{cite journal | last=Wall | first=Charles R. | title=New unitary perfect numbers have at least nine odd components | journal=Fibonacci Quarterly | volume=26 | number=4 | pages=312–317 | year=1988 | issn=0015-0517 | mr=967649 | zbl=0657.10003 }}

{{cite book|author=Richard K. Guy|authorlink=Richard K. Guy|title=Unsolved Problems in Number Theory|publisher=Springer-Verlag|year=2004|isbn=0-387-20860-7 | pages=84–86}} Section B3.
{{cite book | title=My Numbers, My Friends: Popular Lectures on Number Theory | authorlink=Paulo Ribenboim | author=Paulo Ribenboim | publisher=Springer-Verlag | year=2000 | isbn=0-387-98911-0 | page=352 }}
{{cite book | editor1-last=Sándor | editor1-first=József | editor2-last=Mitrinović | editor2-first=Dragoslav S. | editor3-last=Crstici |editor3-first=Borislav | title=Handbook of number theory I | location=Dordrecht | publisher=Springer-Verlag | year=2006 | isbn=1-4020-4215-9 | zbl=1151.11300 }}
{{cite book | last1=Sándor | first1=Jozsef | last2=Crstici | first2=Borislav | title=Handbook of number theory II | location=Dordrecht | publisher=Kluwer Academic | year=2004 | isbn=1-4020-2546-7 | zbl=1079.11001 }}

1 : Integer sequences

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