# Hyperperfect number

In number theory, a k-hyperperfect number is a natural number n for which the equality $n=1+k(\sigma (n)-n-1)$ holds, where σ(n) is the divisor function (i.e., the sum of all positive divisors of n). A hyperperfect number is a k-hyperperfect number for some integer k. Hyperperfect numbers generalize perfect numbers, which are 1-hyperperfect.
The first few numbers in the sequence of k-hyperperfect numbers are 6, 21, 28, 301, 325, 496, 697, ... (sequence A034897 in the OEIS), with the corresponding values of k being 1, 2, 1, 6, 3, 1, 12, ... (sequence A034898 in the OEIS). The first few k-hyperperfect numbers that are not perfect are 21, 301, 325, 697, 1333, ... (sequence A007592 in the OEIS).

## List of hyperperfect numbers

The following table lists the first few k-hyperperfect numbers for some values of k, together with the sequence number in the On-Line Encyclopedia of Integer Sequences (OEIS) of the sequence of k-hyperperfect numbers:
List of some known k-hyperperfect numbers :
| k | k-hyperperfect numbers | OEIS |
| 1 | 6, 28, 496, 8128, 33550336, ... | OEIS: A000396 |
| 2 | 21, 2133, 19521, 176661, 129127041, ... | OEIS: A007593 |
| 3 | 325, ... | |
| 4 | 1950625, 1220640625, ... | |
| 6 | 301, 16513, 60110701, 1977225901, ... | OEIS: A028499 |
| 10 | 159841, ... | |
| 11 | 10693, ... | |
| 12 | 697, 2041, 1570153, 62722153, 10604156641, 13544168521, ... | OEIS: A028500 |
| 18 | 1333, 1909, 2469601, 893748277, ... | OEIS: A028501 |
| 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, ... | OEIS: A028502 |
| 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, ... | OEIS: A034916 |
| 55848 | 15166641361, 44783952721, 67623550801, ... | |
| 67782 | 18407557741, 18444431149, 34939858669, ... | |
| 92568 | 50611924273, 64781493169, 84213367729, ... | |
| 100932 | 50969246953, 53192980777, 82145123113, ... | |
It can be shown that if k > 1 is an odd integer and $p={\tfrac {3k+1}{2}}$ and $q=3k+4$ are prime numbers, then ⁠$p^{2}q$⁠ is k-hyperperfect; Judson S. McCranie has conjectured in 2000 that all k-hyperperfect numbers for odd k > 1 are of this form, but the hypothesis has not been proven so far. Furthermore, it can be proven that if p ≠ q are odd primes and k is an integer such that $k(p+q)=pq-1,$ then pq is k-hyperperfect.
It is also possible to show that if k > 0 and $p=k+1$ is prime, then for all i > 1 such that $q=p^{i}-p+1$ is prime, $n=p^{i-1}q$ is k-hyperperfect. The following table lists known values of k and corresponding values of i for which n is k-hyperperfect:
Values of i for which n is k-hyperperfect :
| k | Values of i | OEIS |
| 16 | 11, 21, 127, 149, 469, ... | OEIS: A034922 |
| 22 | 17, 61, 445, ... | |
| 28 | 33, 89, 101, ... | |
| 36 | 67, 95, 341, ... | |
| 42 | 4, 6, 42, 64, 65, ... | OEIS: A034923 |
| 46 | 5, 11, 13, 53, 115, ... | OEIS: A034924 |
| 52 | 21, 173, ... | |
| 58 | 11, 117, ... | |
| 72 | 21, 49, ... | |
| 88 | 9, 41, 51, 109, 483, ... | OEIS: A034925 |
| 96 | 6, 11, 34, ... | |
| 100 | 3, 7, 9, 19, 29, 99, 145, ... | OEIS: A034926 |