In number theory, a -hyperperfect number is a natural number for which the equality
n=1+k(\sigma(n)-n-1)
The first few numbers in the sequence of -hyperperfect numbers are, with the corresponding values of being . The first few -hyperperfect numbers that are not perfect are .
The following table lists the first few -hyperperfect numbers for some values of, together with the sequence number in the On-Line Encyclopedia of Integer Sequences (OEIS) of the sequence of -hyperperfect numbers:
| -hyperperfect numbers | OEIS | ||
|---|---|---|---|
| 1 | 6, 28, 496, 8128, 33550336, ... | ||
| 2 | 21, 2133, 19521, 176661, 129127041, ... | ||
| 3 | 325, ... | ||
| 4 | 1950625, 1220640625, ... | ||
| 6 | 301, 16513, 60110701, 1977225901, ... | ||
| 10 | 159841, ... | ||
| 11 | 10693, ... | ||
| 12 | 697, 2041, 1570153, 62722153, 10604156641, 13544168521, ... | ||
| 18 | 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 | 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 | 4660241041, 7220722321, 12994506001, 52929885457, 60771359377, ... | ||
| 55848 | 15166641361, 44783952721, 67623550801, ... | ||
| 67782 | 18407557741, 18444431149, 34939858669, ... | ||
| 92568 | 50611924273, 64781493169, 84213367729, ... | ||
| 100932 | 50969246953, 53192980777, 82145123113, ... |
It can be shown that if is an odd integer and
p=\tfrac{3k+1}{2}
q=3k+4
k(p+q)=pq-1,
It is also possible to show that if and
p=k+1
q=pi-p+1
n=pi-1q
| Values of | OEIS | ||
|---|---|---|---|
| 16 | 11, 21, 127, 149, 469, ... | ||
| 22 | 17, 61, 445, ... | ||
| 28 | 33, 89, 101, ... | ||
| 36 | 67, 95, 341, ... | ||
| 42 | 4, 6, 42, 64, 65, ... | ||
| 46 | 5, 11, 13, 53, 115, ... | ||
| 52 | 21, 173, ... | ||
| 58 | 11, 117, ... | ||
| 72 | 21, 49, ... | ||
| 88 | 9, 41, 51, 109, 483, ... | ||
| 96 | 6, 11, 34, ... | ||
| 100 | 3, 7, 9, 19, 29, 99, 145, ... |