# 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).

## Contents

## List of hyperperfect numbers[edit]

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* = (3*k* + 1) / 2 and *q* = 3*k* + 4 are prime numbers, then *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 *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* = *p*^{i} − *p* + 1 is prime, *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:

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[edit]

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

δ_{k}(n) = n(k+1) +(k-1) –kσ(n)

A number n is said to be **k-hyperdeficient** if δ_{k}(n) > 0.

Note that for k=1 one gets δ_{1}(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, δ_{k}(n) = 0.

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

## References[edit]

- 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[edit]

### Articles[edit]

- 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[edit]

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