| 词条 | Quasiperfect number |
| 释义 |
In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the divisor function σ(n)) is equal to 2n + 1. Equivalently, n is the sum of its non-trivial divisors (that is, its divisors excluding 1 and n). No quasiperfect numbers have been found so far. The quasiperfect numbers are the abundant numbers of minimal abundance (which is 1). TheoremsIf a quasiperfect number exists, it must be an odd square number greater than 1035 and have at least seven distinct prime factors.[1] RelatedNumbers do exist where the sum of all the divisors σ(n) is equal to 2n + 2: 20, 104, 464, 650, 1952, 130304, 522752 ... {{OEIS|A088831}}. Many of these numbers are of the form 2n−1(2n − 3) where 2n − 3 is prime (instead of 2n − 1 with perfect numbers). In addition, numbers exist where the sum of all the divisors σ(n) is equal to 2n - 1, such as the powers of 2. Betrothed numbers relate to quasiperfect numbers like amicable numbers relate to perfect numbers. Notes1. ^{{cite journal|last1=Hagis|first1=Peter |last2=Cohen|first2=Graeme L.|title=Some results concerning quasiperfect numbers|journal=J. Austral. Math. Soc. Ser. A|volume=33|year=1982|pages=275–286|doi=10.1017/S1446788700018401|issue=2|mr=0668448}} References
|first2=H. |last2=Abbott |first3=C. |last3=Aull |first4=D. |last4=Suryanarayana |title=Quasiperfect numbers |journal=Acta Arith. |year=1973 |volume=22 |pages=439–447 |mr=0316368 |url=http://matwbn.icm.edu.pl/ksiazki/aa/aa22/aa2245.pdf |doi=10.4064/aa-22-4-439-447 }}
| last=Kishore | first=Masao | title=Odd integers N with five distinct prime factors for which 2−10−12 < σ(N)/N < 2+10−12 | journal=Mathematics of Computation | volume=32 | pages=303–309 | year=1978 | issn=0025-5718 | zbl=0376.10005 | mr=0485658 | url=http://www.ams.org/journals/mcom/1978-32-141/S0025-5718-1978-0485658-X/S0025-5718-1978-0485658-X.pdf | doi=10.2307/2006281 }}
|last1=Cohen|title= On odd perfect numbers (ii), multiperfect numbers and quasiperfect numbers |year=1980 |journal=J. Austral. Math. Soc., Ser. A |volume=29 |pages=369–384 |doi=10.1017/S1446788700021376 | mr=0569525 | zbl=0425.10005 | issn=0263-6115
| author=James J. Tattersall | title=Elementary number theory in nine chapters | publisher=Cambridge University Press | isbn=0-521-58531-7 | year=1999 | pages=147 | zbl=0958.11001 }}
| last = Guy | first = Richard | authorlink = Richard K. Guy | year = 2004 | title = Unsolved Problems in Number Theory, third edition |page=74 | publisher = Springer-Verlag | isbn=0-387-20860-7 }}
| 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 | pages=109–110 }}{{Divisor classes}}{{Classes of natural numbers}}{{numtheory-stub}} 3 : Divisor function|Integer sequences|Unsolved problems in mathematics |
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