Hyperperfect number
In number theory, a -hyperperfect number is a natural number for which the equality holds, where is the divisor function. A hyperperfect number is a -hyperperfect number for some integer. Hyperperfect numbers generalize perfect numbers, which are 1-hyperperfect.
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 .
List of hyperperfect numbers
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 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 numbers|odd] integer and and are prime numbers, then is -hyperperfect; Judson S. McCranie has conjectured in 2000 that all -hyperperfect numbers for odd are of this form, but the hypothesis has not been proven so far. Furthermore, it can be proven that if are odd primes and is an integer such that then is -hyperperfect.
It is also possible to show that if and is prime, then for all such that is prime, is -hyperperfect. The following table lists known values of and corresponding values of for which is -hyperperfect:
| Values of | OEIS | |
| 2 | 2, 4, 5, 6, 9, 22, 37, 41, 90, 102, 105, 317, 520, 541, 561, 648, 780, 786, 957, 1353, 2224, 2521, 6184, 7989, 8890, 19217, 20746, 31722, 37056, 69581, 195430, 225922, 506233, 761457, 1180181,... | |
| 4 | 5, 7, 15, 47, 81, 115, 267, 285, 7641, 19089, 25831, 32115, 59811, 70155, 178715,... | |
| 6 | 2, 3, 6, 9, 21, 25, 33, 49, 54, 133, 245, 255, 318, 1023, 1486, 3334, 6821, 8555, 11605, 42502, 44409, 90291, 92511, 140303,... | |
| 10 | 3, 17, 23, 79, 273, 2185, 4087, 5855, 17151,..., 79133,... | |
| 12 | 2, 4, 5, 6, 13, 24, 64, 133, 268, 744, 952, 1261, 5794, 11833,... | |
| 16 | 11, 21, 127, 149, 469, 2019, 13953, 21689, 25679,..., 81417,... | |
| 18 | 3, 4, 5, 7, 10, 12, 22, 52, 65, 125, 197, 267, 335, 348, 412, 1666, 1705, 3318, 11271,..., 37074,..., 61980,..., 69025,... | |
| 22 | 17, 61, 445, 4381, 15041, 17569,... | |
| 28 | 33, 89, 101, 2439, 4605, 5905, 21193, 24183,... | |
| 30 | 3, 5, 29, 103, 106, 174, 615, 954, 1378, 5622, 6258, 8493, 13639, 14891,..., 26243,..., 31835,..., 59713,..., 78759,... | |
| 36 | 67, 95, 341, 577, 2651, 11761,... | |
| 40 | 3, 5, 55, 161, 197, 1697, 11991, 32295, 57783,... | |
| 42 | 4, 6, 42, 64, 65, 1017, 3390, 3894, 8904, 12976, 63177,... | |
| 46 | 5, 11, 13, 53, 115, 899, 2287, 47667,... | |
| 52 | 21, 173, 2153, 11793,... | |
| 58 | 11, 117, 21351,... | |
| 60 | 5, 13, 24, 42, 81, 112, 2592, 7609, 13054, 23088, 46427,... | |
| 66 | 2, 65, 345, 373, 2073, 4158, 4839, 39701,... | |
| 70 | 3019, 19719,... | |
| 72 | 21, 49, 1744, 2901, 6918, 7320,... | |
| 78 | 2, 4, 16, 29, 47, 142, 352, 4051, 9587,... | |
| 82 | 965, 2421, 12377,... | |
| 88 | 9, 41, 51, 109, 483, 42211,... | |
| 96 | 6, 11, 34, 12239, 12503, 19937,... | |
| 100 | 3, 7, 9, 19, 29, 99, 145, 623, 3001, 6225,..., 23163,... | |
| 102 | 5, 17, 18, 40, 42, 45, 3616, 10441, 13192, 36005, 47825,... | |
| 106 | 7, 745, 3031,..., 53125,... | |
| 108 | 4, 12, 19, 33, 88, 112, 225, 528, 870, 1936, 54683,... |
Articles
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Books
- Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002,