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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 OEIS), with the corresponding values of k being 1, 2, 1, 6, 3, 1, 12, ... (sequence A034898 in OEIS). The first few k-hyperperfect numbers that are not perfect are 21, 301, 325, 697, 1333, ... (sequence A007592 in 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 | A000396 | 6, 28, 496, 8128, 33550336, ... |
| 2 | A007593 | 21, 2133, 19521, 176661, 129127041, ... |
| 3 | 325, ... | |
| 4 | 1950625, 1220640625, ... | |
| 6 | A028499 | 301, 16513, 60110701, 1977225901, ... |
| 10 | 159841, ... | |
| 11 | 10693, ... | |
| 12 | A028500 | 697, 2041, 1570153, 62722153, 10604156641, 13544168521, ... |
| 18 | A028501 | 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 | A028502 | 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 | A034916 | 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 | A034922 | 11, 21, 127, 149, 469, ... |
| 22 | 17, 61, 445, ... | |
| 28 | 33, 89, 101, ... | |
| 36 | 67, 95, 341, ... | |
| 42 | A034923 | 4, 6, 42, 64, 65, ... |
| 46 | A034924 | 5, 11, 13, 53, 115, ... |
| 52 | 21, 173, ... | |
| 58 | 11, 117, ... | |
| 72 | 21, 49, ... | |
| 88 | A034925 | 9, 41, 51, 109, 483, ... |
| 96 | 6, 11, 34, ... | |
| 100 | A034926 | 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, -∞<k<∞, define the k-hyperdeficiency (or simply the hyperdeficiency) for the number n as
δk(n) = n(k+1) +(k-1) –kσ(n)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.
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.
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 595066, 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
MathWorld: Hyperperfect number
A long list of hyperperfect numbers under Data
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