Hyperperfect number
In number theory, a Template:Mvar-hyperperfect number is a natural number Template:Mvar for which the equality holds, where Template:Math is the divisor function (i.e., the sum of all positive divisors of Template:Mvar). A hyperperfect number is a Template:Mvar-hyperperfect number for some integer Template:Mvar. Hyperperfect numbers generalize perfect numbers, which are 1-hyperperfect.[1]
The first few numbers in the sequence of Template:Mvar-hyperperfect numbers are 6, 21, 28, 301, 325, 496, 697, ... (sequence A034897 in the OEIS), with the corresponding values of Template:Mvar being 1, 2, 1, 6, 3, 1, 12, ... (sequence A034898 in the OEIS). The first few Template:Mvar-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 Template:Mvar-hyperperfect numbers for some values of Template:Mvar, together with the sequence number in the On-Line Encyclopedia of Integer Sequences (OEIS) of the sequence of Template:Mvar-hyperperfect numbers:
| Template:Mvar | Template:Mvar-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 Template:Math is an odd integer and and are prime numbers, then Template:Tmath is Template:Mvar-hyperperfect; Judson S. McCranie has conjectured in 2000 that all Template:Mvar-hyperperfect numbers for odd Template:Math are of this form, but the hypothesis has not been proven so far. Furthermore, it can be proven that if Template:Math are odd primes and Template:Mvar is an integer such that then Template:Mvar is Template:Mvar-hyperperfect.
It is also possible to show that if Template:Math and is prime, then for all Template:Math such that is prime, is Template:Mvar-hyperperfect. The following table lists known values of Template:Mvar and corresponding values of Template:Mvar for which Template:Mvar is Template:Mvar-hyperperfect:
| Template:Mvar | Values of Template:Mvar | 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, ... | OEIS: A014224 |
| 4 | 5, 7, 15, 47, 81, 115, 267, 285, 7641, 19089, 25831, 32115, 59811, 70155, 178715, ... | OEIS: A059613 |
| 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, ... | OEIS: A191469 |
| 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, ... | OEIS: A034922 |
| 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, ... | OEIS: A034923 |
| 46 | 5, 11, 13, 53, 115, 899, 2287, 47667, ... | OEIS: A034924 |
| 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, ... | OEIS: A034925 |
| 96 | 6, 11, 34, 12239, 12503, 19937, ... | |
| 100 | 3, 7, 9, 19, 29, 99, 145, 623, 3001, 6225, ..., 23163, ... | OEIS: A034926 |
| 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, ... |
References
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Further reading
Articles
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Books
- Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, Template:ISBN (p. 114-134)
External links
Template:Divisor classes Template:Classes of natural numbers
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