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词条 Multiply perfect number
释义

  1. Smallest k-perfect numbers

  2. Properties

  3. Specific values of k

     Perfect numbers  Triperfect numbers 

  4. References

     Sources 

  5. External links

In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number.

For a given natural number k, a number n is called k-perfect (or k-fold perfect) if and only if the sum of all positive divisors of n (the divisor function, σ(n)) is equal to kn; a number is thus perfect if and only if it is 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number. As of 2014, k-perfect numbers are known for each value of k up to 11.[1]

It can be proven that:

  • For a given prime number p, if n is p-perfect and p does not divide n, then pn is (p+1)-perfect. This implies that an integer n is a 3-perfect number divisible by 2 but not by 4, if and only if n/2 is an odd perfect number, of which none are known.
  • If 3n is 4k-perfect and 3 does not divide n, then n is 3k-perfect.

An open question is whether all k-perfect numbers are divisible by k!, where "!" is the factorial.

Smallest k-perfect numbers

The following table gives an overview of the smallest k-perfect numbers for k ≤ 11 {{OEIS|A007539}}:

k Smallest k-perfect number Factors Found by
1 1 ancient
2 6 2 × 3ancient
3 120 23 × 3 × 5 ancient
4 30240 25 × 33 × 5 × 7 René Descartes, circa 1638
5 14182439040 27 × 34 × 5 × 7 × 112 × 17 × 19 René Descartes, circa 1638
6 154345556085770649600 (21 digits) 215 × 35 × 52 × 72 × 11 × 13 × 17 × 19 × 31 × 43 × 257 Robert Daniel Carmichael, 1907
7 141310897947438348259849402738485523264343544818565120000 (57 digits) 232 × 311 × 54 × 75 × 112 × 132 × 17 × 193 × 23 × 31 × 37 × 43 × 61 × 71 × 73 × 89 × 181 × 2141 × 599479 TE Mason, 1911
8 826809968707776137289924194863596289350194388329245554884393242141388447
6391773708366277840568053624227289196057256213348352000000000 (133 digits) 		262 × 315 × 59 × 77 × 113 × 133 × 172 × 19 × 23 × 29 × 312 × 37 × 41 × 43 × 53 × 612 × 712 × 73 × 83 × 89 × 972 × 127 × 193 × 283 × 307 × 317 × 331 × 337 × 487 × 5212 × 601 × 1201 × 1279 × 2557 × 3169 × 5113 × 92737 × 649657 Stephen F. Gretton, 1990[1]
9 561308081837371589999987...415685343739904000000000 (287 digits) 2104 × 343 × 59 × 712 × 116 × 134 × 17 × 194 × 232 × 29 × 314 × 373 × 412 × 432 × 472 × 53 × 59 × 61 × 67 × 713 × 73 × 792 × 83 × 89 × 97 × 1032 × 107 × 127 × 1312 × 1372 × 1512 × 191 × 211 × 241 × 331 × 337 × 431 × 521 × 547 × 631 × 661 × 683 × 709 × 911 × 1093 × 1301 × 1723 × 2521 × 3067 × 3571 × 3851 × 5501 × 6829 × 6911 × 8647 × 17293 × 17351 × 29191 × 30941 × 45319 × 106681 × 110563 × 122921 × 152041 × 570461 × 16148168401 Fred Helenius, 1995[1]
10 448565429898310924320164...000000000000000000000000 (639 digits) 2175 × 369 × 529 × 718 × 1119 × 138 × 179 × 197 × 239 × 293 × 318 × 372 × 414 × 434 × 474 × 533 × 59 × 615 × 674 × 714 × 732 × 79 × 83 × 89 × 97 × 1013 × 1032 × 1072 × 109 × 113 × 1272 × 1312 × 139 × 149 × 151 × 163 × 179 × 1812 × 191 × 197 × 199 × 2113 × 223 × 239 × 257 × 271 × 281 × 307 × 331 × 337 × 3532 × 367 × 373 × 397 × 419 × 421 × 521 × 523 × 5472 × 613 × 683 × 761 × 827 × 971 × 991 × 1093 × 1741 × 1801 × 2113 × 2221 × 2237 × 2437 × 2551 × 2851 × 3221 × 3571 × 3637 × 3833 × 4339 × 5101 × 5419 × 6577 × 6709 × 7621 × 7699 × 8269 × 8647 × 11093 × 13421 × 13441 × 14621 × 17293 × 26417 × 26881 × 31723 × 44371 × 81343 × 88741 × 114577 × 160967 × 189799 × 229153 × 292561 × 579281 × 581173 × 583367 × 1609669 × 3500201 × 119782433 × 212601841 × 2664097031 × 2931542417 × 43872038849 × 374857981681 × 4534166740403 George Woltman, 2013[1]
11 251850413483992918774837...000000000000000000000000 (1907 digits) 2468 × 3140 × 566 × 749 × 1140 × 1331 × 1711 × 1912 × 239 × 297 × 3111 × 378 × 415 × 433 × 473 × 534 × 593 × 612 × 674 × 714 × 733 × 79 × 832 × 89 × 974 × 1014 × 1033 × 1093 × 1132 × 1273 × 1313 × 1372 × 1392 × 1492 × 151 × 1572 × 163 × 167 × 173 × 181 × 191 × 1932 × 197 × 199 × 2113 × 223 × 227 × 2292 × 239 × 251 × 257 × 263 × 2693 × 271 × 2812 × 293 × 3073 × 313 × 317 × 331 × 347 × 349 × 367 × 373 × 397 × 401 × 419 × 421 × 431 × 4432 × 449 × 457 × 461 × 467 × 491 × 4992 × 541 × 547 × 569 × 571 × 599 × 607 × 613 × 647 × 691 × 701 × 719 × 727 × 761 × 827 × 853 × 937 × 967 × 991 × 997 × 1013 × 1061 × 1087 × 1171 × 1213 × 1223 × 1231 × 1279 × 1381 × 1399 × 1433 × 1609 × 1613 × 1619 × 1723 × 1741 × 1783 × 1873 × 1933 × 1979 × 2081 × 2089 × 2221 × 2357 × 2551 × 2657 × 2671 × 2749 × 2791 × 2801 × 2803 × 3331 × 3433 × 4051 × 4177 × 4231 × 5581 × 5653 × 5839 × 6661 × 7237 × 7699 × 8081 × 8101 × 8269 × 8581 × 8941 × 10501 × 11833 × 12583 × 12941 × 13441 × 14281 × 15053 × 17929 × 19181 × 20809 × 21997 × 23063 × 23971 × 26399 × 26881 × 27061 × 28099 × 29251 × 32051 × 32059 × 32323 × 33347 × 33637 × 36373 × 38197 × 41617 × 51853 × 62011 × 67927 × 73547 × 77081 × 83233 × 92251 × 93253 × 124021 × 133387 × 141311 × 175433 × 248041 × 256471 × 262321 × 292561 × 338753 × 353641 × 441281 × 449653 × 509221 × 511801 × 540079 × 639083 × 696607 × 746023 × 922561 × 1095551 × 1401943 × 1412753 × 1428127 × 1984327 × 2556331 × 5112661 × 5714803 × 7450297 × 8334721 × 10715147 × 14091139 × 14092193 × 18739907 × 19270249 × 29866451 × 96656723 × 133338869 × 193707721 × 283763713 × 407865361 × 700116563 × 795217607 × 3035864933 × 3336809191 × 35061928679 × 143881112839 × 161969595577 × 287762225677 × 761838257287 × 840139875599 × 2031161085853 × 2454335007529 × 2765759031089 × 31280679788951 × 75364676329903 × 901563572369231 × 2169378653672701 × 4764764439424783 × 70321958644800017 × 79787519018560501 × 702022478271339803 × 1839633098314450447 × 165301473942399079669 × 604088623657497125653141 × 160014034995323841360748039 × 25922273669242462300441182317 × 15428152323948966909689390436420781 × 420391294797275951862132367930818883361 × 23735410086474640244277823338130677687887 × 628683935022908831926019116410056880219316806841500141982334538232031397827230330241 George Woltman, 2001[1]

For example, 120 is 3-perfect because the sum of the divisors of 120 is

1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120 = 360 = 3 × 120.

Properties

  • The number of multiperfect numbers less than X is for all positive ε.[2]
  • The only known odd multiply perfect number is 1.{{Citation needed|date=June 2015}}

Specific values of k

Perfect numbers

{{main|Perfect number}}

A number n with σ(n) = 2n is perfect.

Triperfect numbers

A number n with σ(n) = 3n is triperfect. An odd triperfect number must exceed 1070, have at least 12 distinct prime factors, the largest exceeding 105.[3]

References

1. ^1 2 3 4 {{cite web |url=http://wwwhomes.uni-bielefeld.de/achim/mpn.html |title=The Multiply Perfect Numbers Page |accessdate=22 January 2014 |first=Achim |last=Flammenkamp}}
2. ^{{harvnb|Sándor|Mitrinović|Crstici|2006|p=105}}
3. ^{{harvnb|Sándor|Mitrinović|Crstici|2006|pp=108–109}}

Sources

  • {{cite journal

|first1=Kevin A.
|last1=Broughan
|first2=Qizhi
|last2=Zhou
|title=Odd multiperfect numbers of abundancy 4
|journal=J. Number Theory
|doi=10.1016/j.jnt.2007.02.001
|year=2008
|mr=2419178
|volume=126
|number=6
|pages=1566–1575
}}
  • {{cite book |last=Guy | first=Richard K. | authorlink=Richard K. Guy | title=Unsolved problems in number theory | publisher=Springer-Verlag |edition=3rd | year=2004 |isbn=978-0-387-20860-2 | zbl=1058.11001 | at=B2 }}
  • {{cite journal | zbl=0612.10006 | last=Kishore | first=Masao | title=Odd triperfect numbers are divisible by twelve distinct prime factors | journal=J. Aust. Math. Soc. Ser. A | volume=42 | issue=2 | pages=173–182 | year=1987 | issn=0263-6115 | doi=10.1017/s1446788700028184}}
  • {{cite journal

|first1=Richard
|last1=Laatsch
|title=Measuring the abundancy of integers
|journal=Mathematics Magazine
|jstor=2690424
|year=1986
|volume=59
|number=2
|pages=84–92
|mr=0835144| issn=0025-570X | zbl=0601.10003
|doi=10.2307/2690424}}
  • {{cite journal

|first1=James G.
|last1=Merickel
|title=Problem 10617 (Divisors of sums of divisors)
|journal=Amer. Math. Monthly
|year=1999
|jstor=2589515
|volume=106
|number=7
|page=693
|mr=1543520
|doi=10.2307/2589515
}}
  • {{cite journal

|first1=Richard F.
|last1=Ryan
|title=A simpler dense proof regarding the abundancy index
|journal=Math. Mag.
|year=2003
|volume=76
|number=4
|pages=299–301
|jstor=3219086
|mr=1573698
}}
  • {{cite book | editor1-last=Sándor | editor1-first=Jozsef | editor2-last=Crstici | editor2-first=Borislav | title=Handbook of number theory II | location=Dordrecht | publisher=Kluwer Academic | year=2004 | isbn=1-4020-2546-7 | pages=32–36 | zbl=1079.11001 |ref=harv}}
  • {{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 |ref=harv}}
  • {{Cite thesis|degree=PhD

|first1= Ronald M.
|last1=Sorli
|title=Algorithms in the study of multiperfect and odd perfect numbers
|year=2003
|hdl=10453/20034
|publisher=University of Technology|location= Sydney
}}
  • {{cite journal

|first1= Paul A.
|last1=Weiner
|title=The abundancy ratio, a measure of perfection
|journal=Math. Mag.
|year=2000
|jstor=2690980
|volume=73
|number=4
|pages=307–310
|mr=1573474
|doi=10.1080/0025570x.2000.11996860
}}

External links

  • The Multiply Perfect Numbers page
  • The Prime Glossary: Multiply perfect numbers
  • {{cite web |last1=Grime |first1=James |title=The Six Triperfect Numbers |url=https://www.youtube.com/watch?v=DhPtIf-hpuU |website=youTube |publisher=Brady Haran |accessdate=29 June 2018 |format=video}}
{{Divisor classes}}{{Classes of natural numbers}}

1 : Integer sequences
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