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| In [[mathematics]], a '''multiply perfect number''' (also called ''multiperfect number'' or ''pluperfect number'') is a generalization of a [[perfect number]].
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| For a given [[natural number]] ''k'', a number ''n'' is called ''k''-perfect (or ''k''-fold perfect) [[if and only if]] the sum of all positive [[divisor]]s of n (the [[divisor function]], ''σ(n)'') is equal to ''kn''; a number is thus [[perfect number|perfect]] [[if and only if]] it is 2-perfect. A number that is ''k''-perfect for a certain ''k'' is called a multiply perfect number. As of July 2004, ''k''-perfect numbers are known for each value of ''k'' up to 11.
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| It can be proven that:
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| * For a given [[prime number]] ''p'', if ''n'' is ''p''-perfect and ''p'' does not divide ''n'', then ''pn'' is (''p''+1)-perfect. This implies that an integer ''n'' is a 3-perfect number divisible by 2 but not by 4, if and only if ''n''/2 is an odd [[perfect number]], of which none are known.
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| * If 3''n'' is 4''k''-perfect and 3 does not divide ''n'', then n is 3''k''-perfect.
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| == Smallest ''k''-perfect numbers ==
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| The following table gives an overview of the smallest ''k''-perfect numbers for ''k'' <= 7 {{OEIS|A007539}}:
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| {| class="wikitable"
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| ! ''k'' !! Smallest ''k''-perfect number !! Found by
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| |-
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| | 1 || [[1 (number)|1]] || ''ancient''
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| |-
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| | 2 || [[6 (number)|6]] || ''ancient''
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| |-
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| | 3 || [[120 (number)|120]] || ''ancient''
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| |-
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| | 4 || 30240 || [[René Descartes]], circa 1638
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| |-
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| | 5 || 14182439040 || René Descartes, circa 1638
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| |-
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| | 6 || 154345556085770649600 || [[Robert Daniel Carmichael]], 1907
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| |-
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| | 7 || 141310897947438348259849402738485523264343544818565120000 || TE Mason, 1911
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| |-
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| | 8 || 2.34111439263306338... *10^161 || [[Paul Poulet]], 1929<ref name=fl>Flammenkamp</ref>
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| |-
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| | 9 || 7.9842491755534198... *10^465 || Fred Helenius<ref name=fl/>
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| |-
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| | 10 || 2.86879876441793479... *10^923 || Ron Sorli<ref name=fl/>
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| |-
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| | 11 || 2.51850413483992918... *10^1906 || [[George Woltman]]<ref name=fl/>
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| |}
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| For example, 120 is 3-perfect because the sum of the divisors of 120 is<br />
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| 1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120 = 360 = 3 × 120.
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| ==Properties==
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| * The number of multiperfect numbers less than ''X'' is <math>o(X^{\epsilon})</math> for all positive ε.<ref name=HBI105>Sándor et al (2006) p.105</ref>
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| ==Specific values of ''k''== | |
| ===Perfect numbers===
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| {{main|Perfect number}}
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| A number ''n'' with σ(''n'') = 2''n'' is '''perfect'''.
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| ===Triperfect numbers===
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| A number ''n'' with σ(''n'') = 3''n'' is '''triperfect'''. An odd triperfect number must exceed 10<sup>70</sup>, have at least 12 distinct prime factors, the largest exceeding 10<sup>5</sup>.<ref>Sandor et al (2006) pp.108-109</ref>
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| ==References==
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| {{reflist}}
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| * {{cite web |url=http://wwwhomes.uni-bielefeld.de/achim/mpn.html |title=The Multiply Perfect Numbers Page |accessdate=22 January 2014 |first=Achim |last=Flammenkamp}}
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| * {{cite journal
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| |first1=Richard
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| |last1=Laatsch
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| |title=Measuring the abundancy of integers
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| |journal=[[Mathematics Magazine]]
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| |jstor=2690424
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| |year=1986
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| |volume=59
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| |number=2
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| |pages=84–92
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| |mr=0835144| issn=0025-570X | zbl=0601.10003 }}
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| * {{cite journal | zbl=0612.10006 | last=Kishore | first=Masao | title=Odd triperfect numbers are divisible by twelve distinct prime factors | journal=J. Aust. Math. Soc. Ser. A | volume=42 | pages=173-182 | year=1987 | issn=0263-6115 }}
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| * {{cite journal
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| |first1=James G.
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| |last1=Merickel
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| |title=Problem 10617 (Divisors of sums of divisors)
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| |journal=Am. Math. Monthly
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| |year=1999
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| |jstor=2589515
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| |volume=106
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| |number=7
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| |page=693
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| |mr=1543520
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| }}
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| * {{cite journal
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| |first1= Paul A.
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| |last1=Weiner
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| |title=The abundancy ratio, a measure of perfection
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| |journal=Math. Mag.
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| |year=2000
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| |jstor=2690980
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| |volume=73
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| |number=4
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| |pages=307–310
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| |mr=1573474
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| }}
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| * {{Citation
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| |first1= Ronald M.
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| |last1=Sorli
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| |title=Algorithms in the study of multiperfect and odd perfect numbers
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| |year=2003
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| |url=http://hdl.handle.net/2100/275
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| }}
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| * {{cite journal
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| |first1=Richard F.
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| |last1=Ryan
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| |title=A simpler dense proof regarding the abundancy index
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| |journal=Math. Mag.
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| |year=2003
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| |volume=76
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| |number=4
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| |pages=299–301
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| |jstor=3219086
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| |mr=1573698
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| }}
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| * {{cite book |last=Guy | first=Richard K. | authorlink=Richard K. Guy | title=Unsolved problems in number theory | publisher=[[Springer-Verlag]] |edition=3rd | year=2004 |isbn=978-0-387-20860-2 | zbl=1058.11001 | at=B2 }}
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| *{{cite journal
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| |first1=Kevin A.
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| |last1=Broughan
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| |first2=Qizhi
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| |last2=Zhou
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| |title=Odd multiperfect numbers of abundancy 4
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| |journal=J. Number Theory
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| |doi=10.1016/j.jnt.2007.02.001
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| |year=2008
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| |mr=2419178
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| |volume=126
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| |number=6
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| |pages=1566–1575
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| }}
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| *{{cite arxiv
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| |first1=Jeffrey
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| |last1=Ward
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| |title=Does ten have a friend?
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| |eprint=0806.1001
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| |mr=2472812
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| }}
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| * {{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 }}
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| * {{cite book | editor1-last=Sándor | editor1-first=Jozsef | editor2-last=Crstici | editor2-first=Borislav | title=Handbook of number theory II | location=Dordrecht | publisher=Kluwer Academic | year=2004 | isbn=1-4020-2546-7 | pages=32–36 | zbl=1079.11001 }}
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| == External links ==
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| * [http://wwwhomes.uni-bielefeld.de/achim/mpn.html The Multiply Perfect Numbers page]
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| * [http://primes.utm.edu/glossary/page.php?sort=MultiplyPerfect The Prime Glossary: Multiply perfect numbers]
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| {{Divisor classes}}
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| {{Classes of natural numbers}}
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| [[Category:Integer sequences]]
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