Multiply perfect number

Demonstration, with Cuisenaire rods, of the 2-perfection of the number 6

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.

Example

The divisors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120. Their sum is 360, which equals $3\times 120$ , so 120 is 3-perfect.

Smallest k-perfect numbers

The following table gives an overview of the smallest k-perfect numbers for k ≤ 11 (sequence A007539 in the OEIS):

k	Smallest k-perfect number	Factors	Found by
1	1		ancient
2	6	2 × 3	ancient
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]

Properties

The number of multiperfect numbers less than X is $o(X^{{\epsilon }})$ for all positive ε.^[2]
The only known odd multiply perfect number is 1.

Specific values of k

Perfect numbers

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 and have at least 12 distinct prime factors, the largest exceeding 10⁵.^[3]

Variations

Unitary multiply perfect numbers

A positive integer n is called a unitary multi k-perfect number if σ^*(n) = kn. A unitary multiply perfect number is simply a unitary multi k-perfect number for some positive integer k. Equivalently, unitary multiply perfect numbers are those n for which n divides σ^*(n). A unitary multi 2-perfect number is naturally called a unitary perfect number. In the case k > 2, no example of a unitary multi k-perfect number is known till now. It is known that if such a number exists, it must be even and greater than 10¹⁰² and must have more than forty four odd prime factors. This problem is probably very difficult to settle.

A divisor d of a positive integer n is called a unitary divisor if gcd(d, n/d) = 1. The concept of unitary divisor was originally due to R. Vaidyanathaswamy (1931) who called such a divisor as block factor. The present terminology is due to E. Cohen (1960). The sum of the (positive) unitary divisors of n is denoted by σ^*(n).

Bi-unitary multiply perfect numbers

A positive integer n is called a bi-unitary multi k-perfect number if σ^**(n) = kn. This concept is due to Peter Hagis (1987). A bi-unitary multiply perfect number is simply a bi-unitary multi k-perfect number for some positive integer k. Equivalently, bi-unitary multiply perfect numbers are those n for which n divides σ^**(n). A bi-unitary multi 2-perfect number is naturally called a bi-unitary perfect number, and a bi-unitary multi 3-perfect number is called a bi-unitary triperfect number.

A divisor d of a positive integer n is called a bi-unitary divisor of n if the greatest common unitary divisor (gcud) of d and n/d equals 1. This concept is due to D. Surynarayana (1972). The sum of the (positive) bi-unitary divisors of n is denoted by σ^**(n).

Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers. Haukkanen and Sitaramaiah (2020) found certain even bi-unitary triperfect numbers.

References

^ ^a ^b ^c ^d ^e Flammenkamp, Achim. "The Multiply Perfect Numbers Page". Retrieved 22 January 2014.
^ Sándor, Mitrinović & Crstici 2006, p. 105
^ Sándor, Mitrinović & Crstici 2006, pp. 108–109

Sources

Broughan, Kevin A.; Zhou, Qizhi (2008). "Odd multiperfect numbers of abundancy 4" (PDF). J. Number Theory. 126 (6): 1566–1575. doi:10.1016/j.jnt.2007.02.001. MR 2419178.
Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. B2. ISBN 978-0-387-20860-2. Zbl 1058.11001.
Haukkanen, Pentti; Sitaramaiah, V. (2020). "Bi-unitary multiperfect numbers, I" (PDF). Notes Number Theory Discrete Math. 26 (1): 93–171. doi:10.7546/nntdm.2020.26.1.93-171.
Haukkanen, Pentti; Sitaramaiah, V. (2020). "Bi-unitary multiperfect numbers, II" (PDF). Notes Number Theory Discrete Math. 26 (2): 1–26. doi:10.7546/nntdm.2020.26.2.1-26.
Haukkanen, Pentti; Sitaramaiah, V. (2020). "Bi-unitary multiperfect numbers, III" (PDF). Notes Number Theory Discrete Math. 26 (3): 33–67. doi:10.7546/nntdm.2020.26.3.33-67.
Haukkanen, Pentti; Sitaramaiah, V. (2020). "Bi-unitary multiperfect numbers, IV(a)". Notes Number Theory Discrete Math. 26 (4): 2–32. doi:10.7546/nntdm.2020.26.4.2-32.
Kishore, Masao (1987). "Odd triperfect numbers are divisible by twelve distinct prime factors". J. Aust. Math. Soc. Ser. A. 42 (2): 173–182. doi:10.1017/s1446788700028184. ISSN 0263-6115. Zbl 0612.10006.
Laatsch, Richard (1986). "Measuring the abundancy of integers". Mathematics Magazine. 59 (2): 84–92. doi:10.2307/2690424. ISSN 0025-570X. JSTOR 2690424. MR 0835144. Zbl 0601.10003.
Merickel, James G. (1999). "Problem 10617 (Divisors of sums of divisors)". Amer. Math. Monthly. 106 (7): 693. doi:10.2307/2589515. JSTOR 2589515. MR 1543520.
Ryan, Richard F. (2003). "A simpler dense proof regarding the abundancy index". Math. Mag. 76 (4): 299–301. JSTOR 3219086. MR 1573698.
Sándor, Jozsef; Crstici, Borislav, eds. (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 32–36. ISBN 1-4020-2546-7. Zbl 1079.11001.
Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. ISBN 1-4020-4215-9. Zbl 1151.11300.
Sorli, Ronald M. (2003). Algorithms in the study of multiperfect and odd perfect numbers (PhD thesis). Sydney: University of Technology. hdl:10453/20034.
Weiner, Paul A. (2000). "The abundancy ratio, a measure of perfection". Math. Mag. 73 (4): 307–310. doi:10.1080/0025570x.2000.11996860. JSTOR 2690980. MR 1573474.

External links

The Multiply Perfect Numbers page
The Prime Glossary: Multiply perfect numbers
Grime, James. "The Six Triperfect Numbers" (video). youTube. Brady Haran. Retrieved 29 June 2018.

[fl-1] Flammenkamp, Achim. "The Multiply Perfect Numbers Page". Retrieved 22 January 2014.

[HBI105-2] Sándor, Mitrinović & Crstici 2006, p. 105

[3] Sándor, Mitrinović & Crstici 2006, pp. 108–109

[1]

[2]

[3]