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hyperperfect number

{{short description|Type of natural number}}

In number theory, a {{mvar|k}}-hyperperfect number is a natural number {{mvar|n}} for which the equality n = 1+k(\sigma(n)-n-1) holds, where {{math|σ(n)}} is the divisor function (i.e., the sum of all positive divisors of {{mvar|n}}). A hyperperfect number is a {{mvar|k}}-hyperperfect number for some integer {{mvar|k}}. Hyperperfect numbers generalize perfect numbers, which are 1-hyperperfect.{{Cite web|last=Weisstein|first=Eric W.|title=Hyperperfect Number|url=https://mathworld.wolfram.com/HyperperfectNumber.html|access-date=2020-08-10|website=mathworld.wolfram.com|language=en}}

The first few numbers in the sequence of {{mvar|k}}-hyperperfect numbers are {{nowrap|6, 21, 28, 301, 325, 496, 697, ...}} {{OEIS|A034897}}, with the corresponding values of {{mvar|k}} being {{nowrap|1, 2, 1, 6, 3, 1, 12, ...}} {{OEIS|id=A034898}}. The first few {{mvar|k}}-hyperperfect numbers that are not perfect are {{nowrap|21, 301, 325, 697, 1333, ...}} {{OEIS|A007592}}.

List of hyperperfect numbers

The following table lists the first few {{mvar|k}}-hyperperfect numbers for some values of {{mvar|k}}, together with the sequence number in the On-Line Encyclopedia of Integer Sequences (OEIS) of the sequence of {{mvar|k}}-hyperperfect numbers:

class="wikitable mw-collapsible mw-collapsed"

|+ class="nowrap" | List of some known {{mvar|k}}-hyperperfect numbers

{{mvar|k}}{{mvar|k}}-hyperperfect numbersOEIS
1

| 6, 28, 496, 8128, 33550336, ... || {{OEIS2C|A000396}}

2

| 21, 2133, 19521, 176661, 129127041, ... || {{OEIS2C|A007593}}

3

| 325, ... ||  

4

| 1950625, 1220640625, ... ||  

6

| 301, 16513, 60110701, 1977225901, ... || {{OEIS2C|A028499}}

10

| 159841, ... ||  

11

| 10693, ... ||  

12

| 697, 2041, 1570153, 62722153, 10604156641, 13544168521, ... || {{OEIS2C|A028500}}

18

| 1333, 1909, 2469601, 893748277, ... || {{OEIS2C|A028501}}

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, ... || {{OEIS2C|A028502}}

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, ... || {{OEIS2C|A034916}}

55848

| 15166641361, 44783952721, 67623550801, ... ||  

67782

| 18407557741, 18444431149, 34939858669, ... ||  

92568

| 50611924273, 64781493169, 84213367729, ... ||  

100932

| 50969246953, 53192980777, 82145123113, ... ||  

It can be shown that if {{math|k > 1}} is an odd integer and p = \tfrac{3k+1}{2} and q = 3k+4 are prime numbers, then {{tmath|p^2q}} is {{mvar|k}}-hyperperfect; Judson S. McCranie has conjectured in 2000 that all {{mvar|k}}-hyperperfect numbers for odd {{math|k > 1}} are of this form, but the hypothesis has not been proven so far. Furthermore, it can be proven that if {{math|p ≠ q}} are odd primes and {{mvar|k}} is an integer such that k(p+q) = pq-1, then {{mvar|pq}} is {{mvar|k}}-hyperperfect.

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

class="wikitable mw-collapsible mw-collapsed"

|+ class="nowrap" | Values of {{mvar|i}} for which {{mvar|n}} is {{mvar|k}}-hyperperfect

{{mvar|k}}Values of {{mvar|i}}OEIS
16

| 11, 21, 127, 149, 469, ... || {{OEIS2C|A034922}}

22

| 17, 61, 445, ... ||  

28

| 33, 89, 101, ... ||  

36

| 67, 95, 341, ... ||  

42

| 4, 6, 42, 64, 65, ... || {{OEIS2C|A034923}}

46

| 5, 11, 13, 53, 115, ... || {{OEIS2C|A034924}}

52

| 21, 173, ... ||  

58

| 11, 117, ... ||  

72

| 21, 49, ... ||  

88

| 9, 41, 51, 109, 483, ... || {{OEIS2C|A034925}}

96

| 6, 11, 34, ... ||  

100

| 3, 7, 9, 19, 29, 99, 145, ... || {{OEIS2C|A034926}}

References

{{reflist}}

  • {{cite book | editor1-last=Sándor | editor1-first=József | editor2-last=Mitrinović | editor2-first=Dragoslav S. | editor3-last=Crstici |editor3-first=Borislav | title=Handbook of number theory I | location=Dordrecht | publisher=Springer-Verlag | year=2006 | isbn=1-4020-4215-9 | zbl=1151.11300 | page=114}}

Further reading

= Articles =

  • {{citation| last1=Minoli | first1=Daniel | first2=Robert | last2=Bear|title=Hyperperfect numbers|journal=Pi Mu Epsilon Journal|volume=6|number=3|date=Fall 1975|pages=153–157}}.
  • {{citation|last1=Minoli | first1=Daniel | title=Sufficient forms for generalized perfect numbers|journal=Annales de la Faculté des Sciences UNAZA|volume=4|number=2|date=Dec 1978|pages=277–302}}.
  • {{citation|last1=Minoli | first1=Daniel | title=Structural issues for hyperperfect numbers|journal=Fibonacci Quarterly|date=Feb 1981|volume=19|number=1|pages=6–14| doi=10.1080/00150517.1981.12430116 }}.
  • {{citation|last1=Minoli | first1=Daniel | title=Issues in non-linear hyperperfect numbers|journal=Mathematics of Computation|volume=34|number=150|date=April 1980|pages=639–645|doi=10.2307/2006107| jstor=2006107 |doi-access=free}}.
  • {{citation|last1=Minoli | first1=Daniel | title=New results for hyperperfect numbers|journal=Abstracts of the American Mathematical Society|date=October 1980|volume=1|number=6|pages=561}}.
  • {{cite book|last1=Minoli | first1=Daniel | first2=W. | last2=Nakamine| title=ICASSP '80. IEEE International Conference on Acoustics, Speech, and Signal Processing | chapter=Mersenne numbers rooted on 3 for number theoretic transforms |year=1980| volume=5 | pages=243–247 | doi=10.1109/ICASSP.1980.1170906 }}.
  • {{citation|first=Judson S. |last=McCranie |title=A study of hyperperfect numbers |journal=Journal of Integer Sequences |volume=3 |year=2000 |page=13 |bibcode=2000JIntS...3...13M |url=http://www.math.uwaterloo.ca/JIS/VOL3/mccranie.html |url-status=dead |archive-url=https://web.archive.org/web/20040405175234/http://www.math.uwaterloo.ca/JIS/VOL3/mccranie.html |archive-date=2004-04-05 }}.
  • {{citation | title=Hyperperfect numbers with three different prime factors | first=Herman J.J. | last=te Riele | author-link=Herman te Riele | journal=Math. Comp. | volume=36 | year=1981 | issue=153 | pages=297–298 | mr=595066 | zbl=0452.10005 | doi=10.1090/s0025-5718-1981-0595066-9| doi-access=free }}.
  • {{citation | last=te Riele | first=Herman J.J. | author-link=Herman te Riele | title=Rules for constructing hyperperfect numbers | zbl=0531.10005 | journal=Fibonacci Q. | volume=22 | pages=50–60 | year=1984 | doi=10.1080/00150517.1984.12429920 }}.

= Books =

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

External links

  • [http://mathworld.wolfram.com/HyperperfectNumber.html MathWorld: Hyperperfect number]
  • [https://web.archive.org/web/20081205065046/http://j.mccranie.home.comcast.net/ A long list of hyperperfect numbers under Data]

{{Divisor classes}}

{{Classes of natural numbers}}

Category:Divisor function

Category:Integer sequences

Category:Perfect numbers