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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, ... (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)

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.
Articles

Daniel Minoli, Robert Bear (Fall 1975), "Hyperperfect numbers", Pi Mu Epsilon Journal 6 (3): 153–157.
Daniel Minoli (Dec 1978), "Sufficient forms for generalized perfect numbers", Annales de la Faculté des Sciences UNAZA 4 (2): 277–302.
Daniel Minoli (Feb. 1981), "Structural issues for hyperperfect numbers", Fibonacci Quarterly 19 (1): 6–14.
Daniel Minoli (April 1980), "Issues in non-linear hyperperfect numbers", Mathematics of Computation 34 (150): 639–645.
Daniel Minoli (October 1980), "New results for hyperperfect numbers", Abstracts of the American Mathematical Society 1 (6): 561.
Daniel Minoli, W. Nakamine (1980), "Mersenne numbers rooted on 3 for number theoretic transforms", International Conference on Acoustics, Speech, and Signal Processing.
Judson S. McCranie (2000), "A study of hyperperfect numbers", Journal of Integer Sequences 3, http://www.math.uwaterloo.ca/JIS/VOL3/mccranie.html.

Books

Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p.114-134)