ハイパー完全数のソースを表示

'''ハイパー完全数'''（ハイパーかんぜんすう、{{Lang-en-short|hyperperfect number}}）とは以下の数式を満たす自然数 ''n'' である。

:<math>n = 1 + k(\sigma(n)-n-1)</math>

ただし''k'' は自然数、''σ''(''n'') は約数関数である。

ハイパー完全数は自然数 ''k'' を用いて ''k'' -ハイパー完全数と表す。ハイパー完全数は完全数を発展させた数で、完全数は 1-ハイパー完全数である。

具体的な ''k'' -ハイパー完全数は [[6]], [[21]], [[28]], [[301]], [[325]], [[496]], [[697]], ...である。({{OEIS|A034897}})

これらの数に対応する ''k'' の値は1, 2, 1, 6, 3, 1, 12, ...である。({{OEIS|A034898}})

またこれらの数の中で完全数ではないが ''k'' -ハイパー完全数である数は[[21]], [[301]], [[325]], 697, 1333, ...である。({{OEIS|A007592}})

==ハイパー完全数のリスト==

以下に ''k'' の値に対応する最初の数個の ''k'' -ハイパー完全数のリストを載せる。

{| class="wikitable"
! ''k'' !! ''k'' -ハイパー完全数 !! OEIS
|-
| {{center|1}} || 6, 28, 496, 8128, 33550336, ...|| {{OEIS|A000396}}
|-
| {{center|2}} || 21, 2133, 19521, 176661, 129127041, ...|| {{OEIS|A007593}} 
|-
| {{center|3}} || 325, ...|| &nbsp; 
|-
| {{center|4}} || 1950625, 1220640625, ...|| {{OEIS|A220290}} 
|-
| {{center|6}} || 301, 16513, 60110701, 1977225901, ...|| {{OEIS|A028499}} 
|-
| {{center|10}} || 159841, ...|| &nbsp; 
|-
| {{center|11}} || 10693, ...|| &nbsp; 
|-
| {{center|12}} || 697, 2041, 1570153, 62722153, 10604156641, 13544168521, ...|| {{OEIS|A028500}} 
|-
| {{center|18}} || 1333, 1909, 2469601, 893748277, ...|| {{OEIS|A028501}} 
|-
| {{center|19}} || 51301, ...|| &nbsp; 
|-
| {{center|30}} || 3901, 28600321, ...|| &nbsp; 
|-
| {{center|31}} || 214273, ...|| &nbsp; 
|-
| {{center|35}} || 306181, ...|| &nbsp; 
|-
| {{center|40}} || 115788961, ...|| &nbsp; 
|-
| {{center|48}} || 26977, 9560844577, ...|| &nbsp; 
|-
| {{center|59}} || 1433701, ...|| &nbsp; 
|-
| {{center|60}} || 24601, ...|| &nbsp; 
|-
| {{center|66}} || 296341, ...|| &nbsp; 
|-
| {{center|75}} || 2924101, ...|| &nbsp; 
|-
| {{center|78}} || 486877, ...|| &nbsp; 
|-
| {{center|91}} || 5199013, ...|| &nbsp; 
|-
| {{center|100}} || 10509080401, ...|| &nbsp; 
|-
| {{center|108}} || 275833, ...|| &nbsp; 
|-
| {{center|126}} || 12161963773, ...|| &nbsp; 
|-
| {{center|132}} || 96361, 130153, 495529, ...|| &nbsp; 
|-
| {{center|136}} || 156276648817, ...|| &nbsp; 
|-
| {{center|138}} || 46727970517, 51886178401, ...|| &nbsp; 
|-
| {{center|140}} || 1118457481, ...|| &nbsp; 
|-
| {{center|168}} || 250321, ...|| &nbsp; 
|-
| {{center|174}} || 7744461466717, ...|| &nbsp; 
|-
| {{center|180}} || 12211188308281, ...|| &nbsp; 
|-
| {{center|190}} || 1167773821, ...|| &nbsp; 
|-
| {{center|192}} || 163201, 137008036993, ...|| &nbsp; 
|-
| {{center|198}} || 1564317613, ...|| &nbsp; 
|-
| {{center|206}} || 626946794653, 54114833564509, ...|| &nbsp; 
|-
| {{center|222}} || 348231627849277, ...|| &nbsp; 
|-
| {{center|228}} || 391854937, 102744892633, 3710434289467, ...|| &nbsp; 
|-
| {{center|252}} || 389593, 1218260233, ...|| &nbsp; 
|-
| {{center|276}} || 72315968283289, ...|| &nbsp; 
|-
| {{center|282}} || 8898807853477, ...|| &nbsp; 
|-
| {{center|296}} || 444574821937, ...|| &nbsp; 
|-
| {{center|342}} || 542413, 26199602893, ...|| &nbsp; 
|-
| {{center|348}} || 66239465233897, ...|| &nbsp; 
|-
| {{center|350}} || 140460782701, ...|| &nbsp; 
|-
| {{center|360}} || 23911458481, ...|| &nbsp; 
|-
| {{center|366}} || 808861, ...|| &nbsp; 
|-
| {{center|372}} || 2469439417, ...|| &nbsp; 
|-
| {{center|396}} || 8432772615433, ...|| &nbsp; 
|-
| {{center|402}} || 8942902453, 813535908179653, ...|| &nbsp; 
|-
| {{center|408}} || 1238906223697, ...|| &nbsp; 
|-
| {{center|414}} || 8062678298557, ...|| &nbsp; 
|-
| {{center|430}} || 124528653669661, ...|| &nbsp; 
|-
| {{center|438}} || 6287557453, ...|| &nbsp; 
|-
| {{center|480}} || 1324790832961, ...|| &nbsp; 
|-
| {{center|522}} || 723378252872773, 106049331638192773, ...|| &nbsp; 
|-
| {{center|546}} || 211125067071829, ...|| &nbsp; 
|-
| {{center|570}} || 1345711391461, 5810517340434661, ...|| &nbsp; 
|-
| {{center|660}} || 13786783637881, ...|| &nbsp; 
|-
| {{center|672}} || 142718568339485377, ...|| &nbsp; 
|-
| {{center|684}} || 154643791177, ...|| &nbsp; 
|-
| {{center|774}} || 8695993590900027, ...|| &nbsp; 
|-
| {{center|810}} || 5646270598021, ...|| &nbsp; 
|-
| {{center|814}} || 31571188513, ...|| &nbsp; 
|-
| {{center|816}} || 31571188513, ...|| &nbsp; 
|-
| {{center|820}} || 1119337766869561, ...|| &nbsp; 
|-
| {{center|968}} || 52335185632753, ...|| &nbsp; 
|-
| {{center|972}} || 289085338292617, ...|| &nbsp; 
|-
| {{center|978}} || 60246544949557, ...|| &nbsp; 
|-
| {{center|1050}} || 64169172901, ...|| &nbsp; 
|-
| {{center|1410}} || 80293806421, ...|| &nbsp; 
|-
| {{center|2772}} || 95295817, 124035913, ...|| {{OEIS|A028502}} 
|-
| {{center|3918}} || 61442077, 217033693, 12059549149, 60174845917, ...|| &nbsp; 
|-
| {{center|9222}} || 404458477, 3426618541, 8983131757, 13027827181, ...|| &nbsp;
|-
| {{center|9828}} || 432373033, 2797540201, 3777981481, 13197765673, ...|| &nbsp;
|-
| {{center|14280}} || 848374801, 2324355601, 4390957201, 16498569361, ...|| &nbsp; 
|-
| {{center|23730}} || 2288948341, 3102982261, 6861054901, 30897836341, ...|| &nbsp; 
|-
| {{center|31752}} || 4660241041, 7220722321, 12994506001, 52929885457, 60771359377, ...|| {{OEIS|A034916}} 
|-
| {{center|55848}} || 15166641361, 44783952721, 67623550801, ...|| &nbsp; 
|-
| {{center|67782}} || 18407557741, 18444431149, 34939858669, ...|| &nbsp; 
|-
| {{center|92568}} || 50611924273, 64781493169, 84213367729, ...|| &nbsp; 
|-
| {{center|100932}} || 50969246953, 53192980777, 82145123113, ...|| &nbsp; 
|}

== 脚注 ==
{{reflist}}

=== 参考文献 ===
* {{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 | page=114}}
* {{citation| last1=Minoli | first1=Daniel | first2=Robert | last2=Bear|title=hyperperfect numbers|journal=Pi Mu Epsilon Journal|volume=6|number=3|date=Fall 1975|pages=153–157}}.
* {{citation|last1=Minoli | first1=Daniel | title=Sufficient forms for generalized perfect numbers|journal=Annales de la Faculté des Sciences [[University of Kinshasa|UNAZA]]|volume=4|number=2|date=Dec 1978|pages=277–302}}.
* {{citation|last1=Minoli | first1=Daniel | title=Structural issues for hyperperfect numbers|journal=Fibonacci Quarterly|date=Feb 1981|volume=19|number=1|pages=6–14}}.
* {{citation|last1=Minoli | first1=Daniel | title=Issues in non-linear hyperperfect numbers|journal=Mathematics of Computation|volume=34|number=150|date=April 1980|pages=639–645|doi=10.2307/2006107}}.
* {{citation|last1=Minoli | first1=Daniel | title=New results for hyperperfect numbers|journal=Abstracts of the American Mathematical Society|date=October 1980|volume=1|number=6|pages=561}}.
* {{citation|last1=Minoli | first1=Daniel | first2=W. | last2=Nakamine|title=Mersenne numbers rooted on 3 for number theoretic transforms|journal=[[International Conference on Acoustics, Speech, and Signal Processing]]|year=1980}}.
* {{citation|first=Judson S.|last=McCranie|title=A study of hyperperfect numbers|journal=Journal of Integer Sequences|volume=3|year=2000|url=http://www.math.uwaterloo.ca/JIS/VOL3/mccranie.html|archiveurl=https://web.archive.org/web/20040405175234/http://www.math.uwaterloo.ca/JIS/VOL3/mccranie.html|archivedate=2004年4月5日|deadurldate=2017年9月}}.
* {{citation | title=hyperperfect numbers with three different prime factors | first=Herman J.J. | last=te Riele | authorlink=Herman te Riele | journal=Math. Comp.  | volume=36 | year=1981 | pages=297–298 | mr=595066 | zbl=0452.10005 | doi=10.1090/s0025-5718-1981-0595066-9}}.
* {{citation | last=te Riele | first=Herman J.J. | authorlink=Herman te Riele | title=Rules for constructing hyperperfect numbers | zbl=0531.10005 | journal=Fibonacci Q. | volume=22 | pages=50-60 | year=1984 }}.
* Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p.&nbsp;114-134)

== 外部リンク ==
* [http://mathworld.wolfram.com/HyperperfectNumber.html MathWorld: Hyperperfect number]
* [https://web.archive.org/web/20081205065046/http://j.mccranie.home.comcast.net/ A long list of hyperperfect numbers under Data]

{{Divisor classes}}
{{DEFAULTSORT:はいぱーかんせんすう}}
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[[Category:整数の類]]