Editing Hyperperfect number (section)

==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}}
|}