{{위키데이터 속성 추적}}
[[자연수]] <math>n</math>이 <math>n = 1 + k(\sigma(n) - n - 1)</math> (단, <math>\sigma(n)</math>는 n의 양의 약수의 합)을 만족했을 때, <math>n</math>을 '''<math>k</math>-초완전수'''(超完全數,hyperfect number)라 부른다. 어떤 수가 [[완전수]]인 것은 그 수가 1-초완전수인 것과 [[동치]]이다.

모든 <math>k</math>-초완전수를 작은 순서대로 나열하면 다음과 같다.
:6, 21, 28, 301, 325, 496, ... ([[OEIS]]의 [https://web.archive.org/web/20010528002725/http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A034897 A034897])

여기에서 각각의 수에 해당하는 <math>k</math> 값은 다음과 같다.
:1, 2, 1, 6, 3, 1, 12, ... (OEIS의 [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A034898 A034898]{{깨진 링크|url=http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A034898 }})

완전수를 제외한 모든 <math>k</math>-초완전수를 나열하면 다음과 같다.
:21, 301, 325, 697, 1333, ... (OEIS의 [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A007592 A007592]{{깨진 링크|url=http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A007592 }})

== 초완전수의 목록 ==
다음 표는 <math>k</math>값에 대한 알려진 <math>k</math>-초완전수를 해당하는 OEIS 수열과 함께 나타내고 있다.

{| class="wikitable"
|- 
! <math>k</math>
! [[OEIS]]
! 알려진 <math>k</math>-초완전수
|-
| 1 || [https://web.archive.org/web/20010715150601/http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000396 A000396] || 6, 28, 496, 8128, 33550336, ...
|-
| 2 || [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A007593 A007593]{{깨진 링크|url=http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A007593 }} || 21, 2133, 19521, 176661, 129127041, ...
|-
| 3 || || 325, ...
|-
| 4 || || 1950625, 1220640625, ...
|-
| 6 || [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A028499 A028499]{{깨진 링크|url=http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A028499 }} || 301, 16513, 60110701, 1977225901, ...
|-
| 10 || || 159841, ...
|-
| 11 || || 10693, ...
|-
| 12 || [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A028500 A028500]{{깨진 링크|url=http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A028500 }} || 697, 2041, 1570153, 62722153, 10604156641, 13544168521, ...
|-
| 18 || [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A028501 A028501]{{깨진 링크|url=http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A028501 }} || 1333, 1909, 2469601, 893748277, ...
|-
| 19 || || 51301, ...
|-
| 30 || || 3901, 28600321, ...
|-
| 31 || || 214273, ...
|-
| 35 || || 306181, ...
|-
| 40 || || 115788961, ...
|-
| 48 || || 26977, 9560844577, ...
|-
| 59 || || 1433701, ...
|-
| 60 || || 24601, ...
|-
| 66 || || 296341, ...
|-
| 75 || || 2924101, ...
|-
| 78 || || 486877, ...
|-
| 91 || || 5199013, ...
|-
| 100 || || 10509080401, ...
|-
| 108 || || 275833, ...
|-
| 126 || || 12161963773, ...
|-
| 132 || || 96361, 130153, 495529, ...
|-
| 136 || || 156276648817, ...
|-
| 138 || || 46727970517, 51886178401, ...
|-
| 140 || || 1118457481, ...
|-
| 168 || || 250321, ...
|-
| 174 || || 7744461466717, ...
|-
| 180 || || 12211188308281, ...
|-
| 190 || || 1167773821, ...
|-
| 192 || || 163201, 137008036993, ...
|-
| 198 || || 1564317613, ...
|-
| 206 || || 626946794653, 54114833564509, ...
|-
| 222 || || 348231627849277, ...
|-
| 228 || || 391854937, 102744892633, 3710434289467, ...
|-
| 252 || || 389593, 1218260233, ...
|-
| 276 || || 72315968283289, ...
|-
| 282 || || 8898807853477, ...
|-
| 296 || || 444574821937, ...
|-
| 342 || || 542413, 26199602893, ...
|-
| 348 || || 66239465233897, ...
|-
| 350 || || 140460782701, ...
|-
| 360 || || 23911458481, ...
|-
| 366 || || 808861, ...
|-
| 372 || || 2469439417, ...
|-
| 396 || || 8432772615433, ...
|-
| 402 || || 8942902453, 813535908179653, ...
|-
| 408 || || 1238906223697, ...
|-
| 414 || || 8062678298557, ...
|-
| 430 || || 124528653669661, ...
|-
| 438 || || 6287557453, ...
|-
| 480 || || 1324790832961, ...
|-
| 522 || || 723378252872773, 106049331638192773, ...
|-
| 546 || || 211125067071829, ...
|-
| 570 || || 1345711391461, 5810517340434661, ...
|-
| 660 || || 13786783637881, ...
|-
| 672 || || 142718568339485377, ...
|-
| 684 || || 154643791177, ...
|-
| 774 || || 8695993590900027, ...
|-
| 810 || || 5646270598021, ...
|-
| 814 || || 31571188513, ...
|-
| 816 || || 31571188513, ...
|-
| 820 || || 1119337766869561, ...
|-
| 968 || || 52335185632753, ...
|-
| 972 || || 289085338292617, ...
|-
| 978 || || 60246544949557, ...
|-
| 1050 || || 64169172901, ...
|-
| 1410 || || 80293806421, ...
|-
| 2772 || [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A028502 A028502]{{깨진 링크|url=http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A028502 }} || 95295817, 124035913, ...
|-
| 3918 || || 61442077, 217033693, 12059549149, 60174845917, ...
|-
| 9222 || || 404458477, 3426618541, 8983131757, 13027827181, ...
|-
| 9828 || || 432373033, 2797540201, 3777981481, 13197765673, ...
|-
| 14280 || || 848374801, 2324355601, 4390957201, 16498569361, ...
|-
| 23730 || || 2288948341, 3102982261, 6861054901, 30897836341, ...
|-
| 31752 || [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A034916 A034916]{{깨진 링크|url=http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A034916 }} || 4660241041, 7220722321, 12994506001, 52929885457, 60771359377, ...
|-
| 55848 || || 15166641361, 44783952721, 67623550801, ...
|-
| 67782 || || 18407557741, 18444431149, 34939858669, ...
|-
| 92568 || || 50611924273, 64781493169, 84213367729, ...
|-
| 100932 || || 50969246953, 53192980777, 82145123113, ...
|}

=== 성질 ===
<math>k</math>가 1보다 큰 홀수이고 <math>p = \frac{3k + 1} 2, \ q = 3k + 4</math>가 모두 [[소수 (수론)|소수]]라면 <math>p^2 q</math>가 <math>k</math>-초완전수라는 것이 증명이 되어 있다. 2000년 Judson S. McCraine이 모든 1보다 큰 홀수 <math>k</math>에 대해 모든 <math>k</math>-초완전수가 이 꼴이라고 추측했지만 아직 증명이 되지 않았다. 서로 다른 홀수인 소수 <math>p</math>와 <math>q</math>가 존재하고 <math>k(p + q) = pq - 1</math>를 만족하는 자연수 <math>k</math>가 있다면 <math>pq</math>는 <math>k</math>-초완전수이다.

<math>k > 0</math>이고 <math>p = k + 1</math>이 소수일 때, <math>q = p^i - p + 1</math>이 소수임을 성립시키는 모든 <math>i</math>에 대해 <math>p^{i - 1} q</math>가 <math>k</math>-초완전수라는 사실도 알려졌다. 아래 테이블은 이를 만족시키는 <math>k</math>와 <math>i</math>의 목록을 나타내고 있다.

{| class="wikitable"
|-
! <math>k</math>
![[OEIS]]
! 가능한 <math>i</math>의 값
|-
| 16 || [https://web.archive.org/web/20010419011547/http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=034922 A034922] || 11, 21, 127, 149, 469, ...
|-
| 22 ||  || 17, 61, 445, ...
|-
| 28 ||  || 33, 89, 101, ...
|-
| 36 ||  || 67, 95, 341, ...
|-
| 42 || [https://web.archive.org/web/20010419012156/http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=034923 A034923] || 4, 6, 42, 64, 65, ...
|-
| 46 || [https://web.archive.org/web/20010419012002/http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=034924 A034924] || 5, 11, 13, 53, 115, ...
|-
| 52 ||  || 21, 173, ...
|-
| 58 ||  || 11, 117, ...
|-
| 72 ||  || 21, 49, ...
|-
| 88 || [https://web.archive.org/web/20010419012642/http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=034925 A034925] || 9, 41, 51, 109, 483, ...
|-
| 96 ||  || 6, 11, 34, ...
|-
| 100 || [https://web.archive.org/web/20010419013218/http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=034926 A034926] || 3, 7, 9, 19, 29, 99, 145, ...
|}

== 같이 보기 ==
* [[완전수]]

== 참고 문헌 ==

=== 논문 ===

* Daniel Minoli, Robert Bear, ''Hyperperfect Numbers'', PME (Pi Mu Epsilon) Journal, University Oklahoma, Fall 1975, pp. 153-157.
* Daniel Minoli, ''Sufficient Forms For Generalized Perfect Numbers'', Ann. Fac. Sciences, Univ. Nation. Zaire, Section Mathem; Vol. 4, No. 2, Dec 1978, pp. 277-302.
* Daniel Minoli, ''Structural Issues For Hyperperfect Numbers'', Fibonacci Quarterly, Feb. 1981, Vol. 19, No. 1, pp. 6-14.
* Daniel Minoli, ''Issues In Non-Linear Hyperperfect Numbers, Mathematics of Computation'', Vol. 34, No. 150, April 1980, pp. 639-645.
* Daniel Minoli, ''New Results For Hyperperfect Numbers'', Abstracts American Math. Soc., October 1980, Issue 6, Vol. 1, pp. 561.
* Daniel Minoli, W. Nakamine, ''Mersenne Numbers Rooted On 3 For Number Theoretic Transforms'', 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.
* Judson S. McCranie, ''A Study of Hyperperfect Numbers'', Journal of Integer Sequences, Vol. 3 (2000), https://web.archive.org/web/20040405175234/http://www.math.uwaterloo.ca/JIS/VOL3/mccranie.html

=== 책 ===

* Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, {{ISBN|0071406158}} (p.114-134)

== 외부 링크 ==
* [http://mathworld.wolfram.com/HyperperfectNumber.html MathWorld: 초완전수] (영문)
{{약수에 따른 정수의 집합}}

[[분류:수론]]
[[분류:정수열]]