Hyperperfect number
In mathematics, a k-hyperperfect number is a natural number n for which the equality n = 1 + k(σ(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:
| k | OEIS | Some known k-hyperperfect numbers |
|---|---|---|
| 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 k > 1 is an odd integer and p = (3k + 1) / 2 and q = 3k + 4 are prime numbers, then p²q 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 = pi − p + 1 is prime, n = pi − 1q is k-hyperperfect. The following table lists known values of k and corresponding values of i for which n is k-hyperperfect:
| k | OEIS | Values of i |
|---|---|---|
| 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, ... |
Hyperdeficiency
The newly introduced mathematical concept of hyperdeficiency is related to the hyperperfect numbers.
Definition (Minoli 2010): For any integer n and for integer k, , define the k-hyperdeficiency (or simply the hyperdeficiency) for the number n as
δk(n) = n(k+1) +(k-1) – kσ(n)
A number n is said to be k-hyperdeficient if δk(n) > 0.
Note that for k=1 one gets δ1(n)= 2n–σ(n), which is the standard traditional definition of deficiency.
Lemma: A number n is k-hyperperfect (including k=1) if and only if the k-hyperdeficiency of n, δk(n) = 0.
Lemma: A number n is k-hyperperfect (including k=1) if and only if for some k, δk-j(n) = -δk+j(n) for at least one j > 0.
References
- Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. p. 114. ISBN 1-4020-4215-9. Zbl 1151.11300.
Further reading
Articles
- Minoli, Daniel; Bear, Robert (Fall 1975), "Hyperperfect numbers", Pi Mu Epsilon Journal, 6 (3): 153–157 .
- Minoli, Daniel (Dec 1978), "Sufficient forms for generalized perfect numbers", Annales de la Faculté des Sciences UNAZA, 4 (2): 277–302 .
- Minoli, Daniel (Feb 1981), "Structural issues for hyperperfect numbers", Fibonacci Quarterly, 19 (1): 6–14 .
- Minoli, Daniel (April 1980), "Issues in non-linear hyperperfect numbers", Mathematics of Computation, 34 (150): 639–645, doi:10.2307/2006107 .
- Minoli, Daniel (October 1980), "New results for hyperperfect numbers", Abstracts of the American Mathematical Society, 1 (6): 561 .
- Minoli, Daniel; Nakamine, W. (1980), "Mersenne numbers rooted on 3 for number theoretic transforms", International Conference on Acoustics, Speech, and Signal Processing .
- McCranie, Judson S. (2000), "A study of hyperperfect numbers", Journal of Integer Sequences, 3, archived from the original on 2004-04-05 .
- te Riele, Herman J.J. (1981), "Hyperperfect numbers with three different prime factors", Math. Comp., 36: 297–298, doi:10.1090/s0025-5718-1981-0595066-9, MR 0595066, Zbl 0452.10005 .
- te Riele, Herman J.J. (1984), "Rules for constructing hyperperfect numbers", Fibonacci Q., 22: 50–60, Zbl 0531.10005 .
Books
- Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p. 114-134)
External links
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