Editing Hyperperfect number

{{short description|Type of natural number}}

In [[number theory]], a '''{{mvar|k}}-hyperperfect number''' is a [[natural number]] {{mvar|n}} for which the equality <math>n = 1+k(\sigma(n)-n-1)</math> holds, where {{math|''σ''(''n'')}} is the [[divisor function]] (i.e., [[Aliquot sum|the sum]] of all positive [[divisor]]s of {{mvar|n}}). A '''hyperperfect number''' is a {{mvar|k}}-hyperperfect number for some integer {{mvar|k}}. Hyperperfect numbers generalize [[perfect number]]s, which are 1-hyperperfect.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Hyperperfect Number|url=https://mathworld.wolfram.com/HyperperfectNumber.html|access-date=2020-08-10|website=mathworld.wolfram.com|language=en}}</ref>

The first few numbers in the sequence of {{mvar|k}}-hyperperfect numbers are {{nowrap|6, 21, 28, 301, 325, 496, 697, ...}} {{OEIS|A034897}}, with the corresponding values of {{mvar|k}} being {{nowrap|1, 2, 1, 6, 3, 1, 12, ...}} {{OEIS|id=A034898}}. The first few {{mvar|k}}-hyperperfect numbers that are not perfect are {{nowrap|21, 301, 325, 697, 1333, ...}} {{OEIS|A007592}}.

==List of hyperperfect numbers==
The following table lists the first few {{mvar|k}}-hyperperfect numbers for some values of {{mvar|k}}, together with the sequence number in the [[On-Line Encyclopedia of Integer Sequences]] (OEIS) of the sequence of {{mvar|k}}-hyperperfect numbers:

{| class="wikitable mw-collapsible mw-collapsed"
|+ class="nowrap" | List of some known {{mvar|k}}-hyperperfect numbers
|-
! {{mvar|k}} !! {{mvar|k}}-hyperperfect numbers !! OEIS
|-
! 1 
| 6, 28, 496, 8128, 33550336, ... || {{OEIS2C|A000396}}
|-
! 2 
| 21, 2133, 19521, 176661, 129127041, ... || {{OEIS2C|A007593}}
|-
! 3 
| 325, ... || &nbsp; 
|-
! 4 
| 1950625, 1220640625, ... || &nbsp; 
|-
! 6 
| 301, 16513, 60110701, 1977225901, ... || {{OEIS2C|A028499}}
|-
! 10 
| 159841, ... || &nbsp; 
|-
! 11 
| 10693, ... || &nbsp; 
|-
! 12 
| 697, 2041, 1570153, 62722153, 10604156641, 13544168521, ... || {{OEIS2C|A028500}}
|-
! 18 
| 1333, 1909, 2469601, 893748277, ... || {{OEIS2C|A028501}}
|-
! 19 
| 51301, ... || &nbsp; 
|-
! 30 
| 3901, 28600321, ... || &nbsp; 
|-
! 31 
| 214273, ... || &nbsp; 
|-
! 35 
| 306181, ... || &nbsp; 
|-
! 40 
| 115788961, ... || &nbsp; 
|-
! 48 
| 26977, 9560844577, ... || &nbsp; 
|-
! 59 
| 1433701, ... || &nbsp; 
|-
! 60 
| 24601, ... || &nbsp; 
|-
! 66 
| 296341, ... || &nbsp; 
|-
! 75 
| 2924101, ... || &nbsp; 
|-
! 78 
| 486877, ... || &nbsp; 
|-
! 91 
| 5199013, ... || &nbsp; 
|-
! 100 
| 10509080401, ... || &nbsp; 
|-
! 108 
| 275833, ... || &nbsp; 
|-
! 126
| 12161963773, ... || &nbsp; 
|-
! 132 
| 96361, 130153, 495529, ... || &nbsp; 
|-
! 136 
| 156276648817, ... || &nbsp; 
|-
! 138 
| 46727970517, 51886178401, ... || &nbsp; 
|-
! 140 
| 1118457481, ... || &nbsp; 
|-
! 168 
| 250321, ... || &nbsp; 
|-
! 174 
| 7744461466717, ... || &nbsp; 
|-
! 180 
| 12211188308281, ... || &nbsp; 
|-
! 190 
| 1167773821, ... || &nbsp; 
|-
! 192 
| 163201, 137008036993, ... || &nbsp; 
|-
! 198 
| 1564317613, ... || &nbsp; 
|-
! 206 
| 626946794653, 54114833564509, ... || &nbsp; 
|-
! 222 
| 348231627849277, ... || &nbsp; 
|-
! 228 
| 391854937, 102744892633, 3710434289467, ... || &nbsp; 
|-
! 252 
| 389593, 1218260233, ... || &nbsp; 
|-
! 276 
| 72315968283289, ... || &nbsp; 
|-
! 282 
| 8898807853477, ... || &nbsp; 
|-
! 296 
| 444574821937, ... || &nbsp; 
|-
! 342 
| 542413, 26199602893, ... || &nbsp; 
|-
! 348 
| 66239465233897, ... || &nbsp; 
|-
! 350 
| 140460782701, ... || &nbsp; 
|-
! 360 
| 23911458481, ... || &nbsp; 
|-
! 366 
| 808861, ... || &nbsp; 
|-
! 372 
| 2469439417, ... || &nbsp; 
|-
! 396 
| 8432772615433, ... || &nbsp; 
|-
! 402 
| 8942902453, 813535908179653, ... || &nbsp; 
|-
! 408 
| 1238906223697, ... || &nbsp; 
|-
! 414 
| 8062678298557, ... || &nbsp; 
|-
! 430 
| 124528653669661, ... || &nbsp; 
|-
! 438 
| 6287557453, ... || &nbsp; 
|-
! 480 
| 1324790832961, ... || &nbsp; 
|-
! 522 
| 723378252872773, 106049331638192773, ... || &nbsp; 
|-
! 546 
| 211125067071829, ... || &nbsp; 
|-
! 570 
| 1345711391461, 5810517340434661, ... || &nbsp; 
|-
! 660 
| 13786783637881, ... || &nbsp; 
|-
! 672 
| 142718568339485377, ... || &nbsp; 
|-
! 684 
| 154643791177, ... || &nbsp; 
|-
! 774 
| 8695993590900027, ... || &nbsp; 
|-
! 810 
| 5646270598021, ... || &nbsp; 
|-
! 814 
| 31571188513, ... || &nbsp; 
|-
! 816 
| 31571188513, ... || &nbsp; 
|-
! 820 
| 1119337766869561, ... || &nbsp; 
|-
! 968 
| 52335185632753, ... || &nbsp; 
|-
! 972 
| 289085338292617, ... || &nbsp; 
|-
! 978 
| 60246544949557, ... || &nbsp; 
|-
! 1050 
| 64169172901, ... || &nbsp; 
|-
! 1410 
| 80293806421, ... || &nbsp; 
|-
! 2772 
| 95295817, 124035913, ... || {{OEIS2C|A028502}} 
|-
! 3918 
| 61442077, 217033693, 12059549149, 60174845917, ... || &nbsp; 
|-
! 9222 
| 404458477, 3426618541, 8983131757, 13027827181, ... || &nbsp; 
|-
! 9828 
| 432373033, 2797540201, 3777981481, 13197765673, ... || &nbsp; 
|-
! 14280 
| 848374801, 2324355601, 4390957201, 16498569361, ... || &nbsp; 
|-
! 23730 
| 2288948341, 3102982261, 6861054901, 30897836341, ... || &nbsp; 
|-
! 31752 
| 4660241041, 7220722321, 12994506001, 52929885457, 60771359377, ... || {{OEIS2C|A034916}}
|-
! 55848 
| 15166641361, 44783952721, 67623550801, ... || &nbsp; 
|-
! 67782 
| 18407557741, 18444431149, 34939858669, ... || &nbsp; 
|-
! 92568 
| 50611924273, 64781493169, 84213367729, ... || &nbsp; 
|-
! 100932 
| 50969246953, 53192980777, 82145123113, ... || &nbsp; 
|}

It can be shown that if {{math|''k'' > 1}} is an [[Even and odd numbers|odd]] [[integer]] and <math>p = \tfrac{3k+1}{2}</math> and <math>q = 3k+4</math> are [[prime number]]s, then {{tmath|p^2q}} is {{mvar|k}}-hyperperfect; Judson S. McCranie has conjectured in 2000 that all {{mvar|k}}-hyperperfect numbers for odd {{math|''k'' > 1}} are of this form, but the hypothesis has not been proven so far. Furthermore, it can be proven that if {{math|''p'' ≠ ''q''}} are odd primes and {{mvar|k}} is an integer such that <math>k(p+q) = pq-1,</math> then {{mvar|pq}} is {{mvar|k}}-hyperperfect.

It is also possible to show that if {{math|''k'' > 0}} and <math>p = k+1</math> is prime, then for all {{math|''i'' > 1}} such that <math>q = p^i - p+1</math> is prime, <math>n = p^{i-1}q</math> is {{mvar|k}}-hyperperfect. The following table lists known values of {{mvar|k}} and corresponding values of {{mvar|i}} for which {{mvar|n}} is {{mvar|k}}-hyperperfect:

{| class="wikitable mw-collapsible mw-collapsed"
|+ class="nowrap" | Values of {{mvar|i}} for which {{mvar|n}} is {{mvar|k}}-hyperperfect
|-
! {{mvar|k}} !! Values of {{mvar|i}} !! OEIS
|-
! 16 
| 11, 21, 127, 149, 469, ... || {{OEIS2C|A034922}}
|-
! 22 
| 17, 61, 445, ... || &nbsp;
|-
! 28 
| 33, 89, 101, ... || &nbsp;
|-
! 36 
| 67, 95, 341, ... || &nbsp;
|-
! 42 
| 4, 6, 42, 64, 65, ... || {{OEIS2C|A034923}}
|-
! 46 
| 5, 11, 13, 53, 115, ... || {{OEIS2C|A034924}}
|-
! 52 
| 21, 173, ... || &nbsp;
|-
! 58 
| 11, 117, ... || &nbsp;
|-
! 72
| 21, 49, ... || &nbsp;
|-
! 88 
| 9, 41, 51, 109, 483, ... || {{OEIS2C|A034925}}
|-
! 96 
| 6, 11, 34, ... || &nbsp;
|-
! 100 
| 3, 7, 9, 19, 29, 99, 145, ... || {{OEIS2C|A034926}}
|}

==References==
{{reflist}}
* {{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}}

== Further reading ==

=== Articles ===

* {{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 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| doi=10.1080/00150517.1981.12430116 }}.
* {{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| jstor=2006107 |doi-access=free}}.
* {{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}}.
* {{cite book|last1=Minoli | first1=Daniel | first2=W. | last2=Nakamine| title=ICASSP '80. IEEE International Conference on Acoustics, Speech, and Signal Processing | chapter=Mersenne numbers rooted on 3 for number theoretic transforms |year=1980| volume=5 | pages=243–247 | doi=10.1109/ICASSP.1980.1170906 }}.
* {{citation|first=Judson S. |last=McCranie |title=A study of hyperperfect numbers |journal=Journal of Integer Sequences |volume=3 |year=2000 |page=13 |bibcode=2000JIntS...3...13M |url=http://www.math.uwaterloo.ca/JIS/VOL3/mccranie.html |url-status=dead |archive-url=https://web.archive.org/web/20040405175234/http://www.math.uwaterloo.ca/JIS/VOL3/mccranie.html |archive-date=2004-04-05 }}.
* {{citation | title=Hyperperfect numbers with three different prime factors | first=Herman J.J. | last=te Riele | author-link=Herman te Riele | journal=Math. Comp.  | volume=36 | year=1981 | issue=153 | pages=297–298 | mr=595066 | zbl=0452.10005 | doi=10.1090/s0025-5718-1981-0595066-9| doi-access=free }}. 
* {{citation | last=te Riele | first=Herman J.J. | author-link=Herman te Riele | title=Rules for constructing hyperperfect numbers | zbl=0531.10005 | journal=Fibonacci Q. | volume=22 | pages=50–60 | year=1984 | doi=10.1080/00150517.1984.12429920 }}.

=== Books ===

* Daniel Minoli, ''Voice over MPLS'', McGraw-Hill, New York, NY, 2002, {{ISBN|0-07-140615-8}} (p.&nbsp;114-134)

== External links ==
* [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}}
{{Classes of natural numbers}}

[[Category:Divisor function]]
[[Category:Integer sequences]]
[[Category:Perfect numbers]]