In number theory, a k-hyperperfect number is a natural number n for which the equality ${\displaystyle n=1+k(\sigma (n)-n-1)}$ 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.[1]

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:

List of some known k-hyperperfect numbers
k k-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 k > 1 is an odd integer and ${\displaystyle p={\tfrac {3k+1}{2))}$ and ${\displaystyle q=3k+4}$ are prime numbers, then ${\displaystyle p^{2}q}$ 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 pq are odd primes and k is an integer such that ${\displaystyle k(p+q)=pq-1,}$ then pq is k-hyperperfect.

It is also possible to show that if k > 0 and ${\displaystyle p=k+1}$ is prime, then for all i > 1 such that ${\displaystyle q=p^{i}-p+1}$ is prime, ${\displaystyle n=p^{i-1}q}$ is k-hyperperfect. The following table lists known values of k and corresponding values of i for which n is k-hyperperfect:

Values of i for which n is k-hyperperfect
k Values of i OEIS
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, ...

There are some Even Numbers which are Hyperperfect for Odd Factors i.e., k * (Sum of Odd Factors except 1 and Itself) + 1 = number. E.g., the first 5 ones include 1300, 271872, 304640, 953344 and 1027584 for k = 3, 349, 353, 837 and 353. All Odd Hyperperfect Numbers are Odd Factor Hyperperfect Numbers as they only have odd factors and have no even factors.

1300 has Factors = 1, 2, 4, 5, 10, 13, 20, 25, 26, 50, 52, 65, 100, 130, 260, 325, 650, 1300

It has Odd Factors except 1 and Itself = 5, 13, 25, 65, 325

Sum of Odd Factors except 1 and Itself = 5 + 13 + 25 + 65 + 325 = 433

1300 - 1 = 1299 and 1299/433 = 3, an Integer[citation needed][clarification needed]

## 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 > 0, define the k-hyperdeficiency (or simply the hyperdeficiency) for the number n as

${\displaystyle \delta _{k}(n)=n(k+1)+(k-1)-k\sigma (n)}$

A number n is said to be k-hyperdeficient if ${\displaystyle \delta _{k}(n)>0.}$

Note that for k = 1 one gets ${\displaystyle \delta _{1}(n)=2n-\sigma (n),}$ 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, ${\displaystyle \delta _{k}(n)=0.}$

Lemma: A number n is k-hyperperfect (including k = 1) if and only if for some k, ${\displaystyle \delta {k-j}(n)=-\delta _{k+j}(n)}$ for at least one j > 0.

## References

1. ^ Weisstein, Eric W. "Hyperperfect Number". mathworld.wolfram.com. Retrieved 2020-08-10.

### 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, JSTOR 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". ICASSP '80. IEEE International Conference on Acoustics, Speech, and Signal Processing. Vol. 5. pp. 243–247. doi:10.1109/ICASSP.1980.1170906..
• McCranie, Judson S. (2000), "A study of hyperperfect numbers", Journal of Integer Sequences, 3: 13, Bibcode:2000JIntS...3...13M, archived from the original on 2004-04-05.
• te Riele, Herman J.J. (1981), "Hyperperfect numbers with three different prime factors", Math. Comp., 36 (153): 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)