multiply perfect number

{{Short description|Number whose divisors add to a multiple of that number}}

File:Multiply perfect number Cuisenaire rods 6.png, of the {{nowrap|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 {{nowrap|k-perfect}} (or {{nowrap|k-fold}} perfect) 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 {{nowrap|2-perfect}}. A number that is {{nowrap|k-perfect}} for a certain k is called a multiply perfect number. As of 2014, {{nowrap|k-perfect}} numbers are known for each value of k up to 11.

It is unknown whether there are any odd multiply perfect numbers other than 1. The first few multiply perfect numbers are:

:1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, ... {{OEIS|A007691}}.

Example

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

which is 3 × 120. Therefore 120 is a {{nowrap|3-perfect}} number.

Smallest known ''k''-perfect numbers

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

class="wikitable"

! k !! Smallest k-perfect number !! Factors !! Found by

11ancient
262 × 3ancient
312023 × 3 × 5ancient
43024025 × 33 × 5 × 7René Descartes, circa 1638
51418243904027 × 34 × 5 × 7 × 112 × 17 × 19René Descartes, circa 1638
6154345556085770649600 (21 digits)215 × 35 × 52 × 72 × 11 × 13 × 17 × 19 × 31 × 43 × 257Robert Daniel Carmichael, 1907
7141310897947438348259849...523264343544818565120000 (57 digits)232 × 311 × 54 × 75 × 112 × 132 × 17 × 193 × 23 × 31 × 37 × 43 × 61 × 71 × 73 × 89 × 181 × 2141 × 599479TE Mason, 1911
8826809968707776137289924...057256213348352000000000 (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 × 649657Stephen F. Gretton, 1990{{cite web |url=http://wwwhomes.uni-bielefeld.de/achim/mpn.html |title=The Multiply Perfect Numbers Page |access-date=22 January 2014 |first=Achim |last=Flammenkamp}}
9561308081837371589999987...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 × 16148168401Fred Helenius, 1995
10448565429898310924320164...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 × 4534166740403George Woltman, 2013
11251850413483992918774837...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 × 628683935022908831926019116410056880219316806841500141982334538232031397827230330241George Woltman, 2001

Properties

It can be proven that:

  • For a given prime number p, if n is {{nowrap|p-perfect}} and p does not divide n, then pn is {{nowrap|(p + 1)-perfect}}. This implies that an integer n is a {{nowrap|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 {{nowrap|4k-perfect}} and 3 does not divide n, then n is {{nowrap|3k-perfect}}.

Odd multiply perfect numbers

{{Unsolved|mathematics|Are there any odd multiply perfect numbers?}}

It is unknown whether there are any odd multiply perfect numbers other than 1. However if an odd {{nowrap|k-perfect}} number n exists where k > 2, then it must satisfy the following conditions:

  • The largest prime factor is ≥ 100129
  • The second largest prime factor is ≥ 1009
  • The third largest prime factor is ≥ 101

Tóth found several numbers that would be odd multiperfect, if one of their factors was a square ({{harvtxt|Tóth|2025}}). An example is 8999757, which would be an odd multiperfect number, if only one of its prime factors, 61, was a square. This is closely related to the concept of Descartes numbers.

Bounds

In little-o notation, the number of multiply perfect numbers less than x is o(x^\varepsilon) for all ε > 0.{{harvnb|Sándor|Mitrinović|Crstici|2006|p=105}}

The number of k-perfect numbers n for nx is less than cx^{c'\log\log\log x/\log\log x}, where c and c' are constants independent of k.

Under the assumption of the Riemann hypothesis, the following inequality is true for all {{nowrap|k-perfect}} numbers n, where k > 3

:\log\log n > k\cdot e^{-\gamma}

where \gamma is Euler's gamma constant. This can be proven using Robin's theorem.

The number of divisors τ(n) of a {{nowrap|k-perfect}} number n satisfies the inequality{{cite arXiv |last=Dagal |first=Keneth Adrian P. |eprint=1309.3527 |title=A Lower Bound for τ(n) for k-Multiperfect Number |class=math.NT |date=2013}}

:\tau(n) > e^{k - \gamma}.

The number of distinct prime factors ω(n) of n satisfies{{harvnb|Sándor|Mitrinović|Crstici|2006|p=106}}

:\omega(n) \ge k^2-1.

If the distinct prime factors of n are p_1, p_2, \ldots, p_r, then:

:r \left(\sqrt[r]{3/2} - 1\right) < \sum_{i=1}^{r} \frac{1}{p_i} < r \left(1 - \sqrt[r]{6/k^2}\right), ~~ \text{if }n\text{ is even}

:r \left(\sqrt[3r]{k^2} - 1\right) < \sum_{i=1}^{r} \frac{1}{p_i} < r \left(1 - \sqrt[r]{8/(k\pi^2)}\right), ~~ \text{if }n\text{ is odd}

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. There are only six known triperfect numbers and these are believed to comprise all such numbers:

: 120, 672, 523776, 459818240, 1476304896, 51001180160 {{OEIS|A005820}}

If there exists an odd perfect number m (a famous open problem) then 2m would be {{nowrap|3-perfect}}, since σ(2m) = σ(2)σ(m) = 3×2m. An odd triperfect number must be a square number exceeding 1070 and have at least 12 distinct prime factors, the largest exceeding 105.{{harvnb|Sándor|Mitrinović|Crstici|2006|pp=108–109}}

Variations

=Unitary multiply perfect numbers=

A similar extension can be made for unitary perfect numbers. A positive integer n is called a unitary multi {{nowrap|k-perfect}} number if σ*(n) = kn where σ*(n) is the sum of its unitary divisors. (A divisor d of a number n is a unitary divisor if d and n/d share no common factors.).

A unitary multiply perfect number is simply a unitary multi {{nowrap|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 {{nowrap|2-perfect}} number is naturally called a unitary perfect number. In the case k > 2, no example of a unitary multi {{nowrap|k-perfect}} number is yet known. 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. 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 first few unitary multiply perfect numbers are:

:1, 6, 60, 90, 87360 {{OEIS|A327158}}

=Bi-unitary multiply perfect numbers=

A positive integer n is called a bi-unitary multi {{nowrap|k-perfect}} number if σ**(n) = kn where σ**(n) is the sum of its bi-unitary divisors. This concept is due to Peter Hagis (1987). A bi-unitary multiply perfect number is simply a bi-unitary multi {{nowrap|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 {{nowrap|2-perfect}} number is naturally called a bi-unitary perfect number, and a bi-unitary multi {{nowrap|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 other than 1. Haukkanen and Sitaramaiah (2020) found all bi-unitary triperfect numbers of the form 2au where 1 ≤ a ≤ 6 and u is odd,{{harvnb|Haukkanen|Sitaramaiah|2020a}}{{harvnb|Haukkanen|Sitaramaiah|2020b}}{{harvnb|Haukkanen|Sitaramaiah|2020c}} and partially the case where a = 7.{{harvnb|Haukkanen|Sitaramaiah|2020d}}

{{harvnb|Haukkanen|Sitaramaiah|2021a}}

Further, they fixed completely the case a = 8.{{harvnb|Haukkanen|Sitaramaiah|2021b}}

Tomohiro Yamada (Determining all biunitary triperfect numbers of a certain form, arXiv:2406.19331 [math.NT], 2024) proved that 2160 = 33 80 is the only biunitary triperfect number of the form 3au where 3 ≤ a and u is not divisible by 3.

The first few bi-unitary multiply perfect numbers are:

:1, 6, 60, 90, 120, 672, 2160, 10080, 22848, 30240 {{OEIS|A189000}}

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See also