[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/multiply-perfect-number-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/multiply-perfect-number-wikipedia\/","headline":"Multiply perfect number – Wikipedia","name":"Multiply perfect number – Wikipedia","description":"Number whose divisors add to a multiple of that number In mathematics, a multiply perfect number (also called multiperfect number","datePublished":"2017-08-16","dateModified":"2017-08-16","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/c\/ca\/Multiply_perfect_number_Cuisenaire_rods_6.png\/220px-Multiply_perfect_number_Cuisenaire_rods_6.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/c\/ca\/Multiply_perfect_number_Cuisenaire_rods_6.png\/220px-Multiply_perfect_number_Cuisenaire_rods_6.png","height":"166","width":"220"},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/multiply-perfect-number-wikipedia\/","wordCount":6800,"articleBody":"Number whose divisors add to a multiple of that number  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 the sum of all positive divisors of n (the divisor function, \u03c3(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 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, … (sequence A007691 in the OEIS).Table of ContentsExample[edit]Smallest known k-perfect numbers[edit]Properties[edit]Odd multiply perfect numbers[edit]Specific values of k[edit]Perfect numbers[edit]Triperfect numbers[edit]Variations[edit]Unitary multiply perfect numbers[edit]Bi-unitary multiply perfect numbers[edit]References[edit]Sources[edit]See also[edit]External links[edit]Example[edit]The sum of the divisors of 120 is1 + 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 + 15 + 20 + 24 + 30 + 40 + 60 + 120 = 360which is 3 \u00d7 120. Therefore 120 is a 3-perfect number.Smallest known k-perfect numbers[edit]The following table gives an overview of the smallest known k-perfect numbers for k \u2264 11 (sequence A007539 in the OEIS):kSmallest known k-perfect numberFactorsFound by11ancient262 \u00d7 3ancient312023 \u00d7 3 \u00d7 5ancient43024025 \u00d7 33 \u00d7 5 \u00d7 7Ren\u00e9 Descartes, circa 163851418243904027 \u00d7 34 \u00d7 5 \u00d7 7 \u00d7 112 \u00d7 17 \u00d7 19Ren\u00e9 Descartes, circa 16386154345556085770649600 (21 digits)215 \u00d7 35 \u00d7 52 \u00d7 72 \u00d7 11 \u00d7 13 \u00d7 17 \u00d7 19 \u00d7 31 \u00d7 43 \u00d7 257Robert Daniel Carmichael, 19077141310897947438348259849402738485523264343544818565120000 (57 digits)232 \u00d7 311 \u00d7 54 \u00d7 75 \u00d7 112 \u00d7 132 \u00d7 17 \u00d7 193 \u00d7 23 \u00d7 31 \u00d7 37 \u00d7 43 \u00d7 61 \u00d7 71 \u00d7 73 \u00d7 89 \u00d7 181 \u00d7 2141 \u00d7 599479TE Mason, 19118826809968707776137289924…057256213348352000000000 (133 digits)262 \u00d7 315 \u00d7 59 \u00d7 77 \u00d7 113 \u00d7 133 \u00d7 172 \u00d7 19 \u00d7 23 \u00d7 29 \u00d7 … \u00d7 487 \u00d7 5212 \u00d7 601 \u00d7 1201 \u00d7 1279 \u00d7 2557 \u00d7 3169 \u00d7 5113 \u00d7 92737 \u00d7 649657 (38 distinct prime factors)Stephen F. Gretton, 1990[1]9561308081837371589999987…415685343739904000000000 (287 digits)2104 \u00d7 343 \u00d7 59 \u00d7 712 \u00d7 116 \u00d7 134 \u00d7 17 \u00d7 194 \u00d7 232 \u00d7 29 \u00d7 … \u00d7 17351 \u00d7 29191 \u00d7 30941 \u00d7 45319 \u00d7 106681 \u00d7 110563 \u00d7 122921 \u00d7 152041 \u00d7 570461 \u00d7 16148168401 (66 distinct prime factors)Fred Helenius, 1995[1]10448565429898310924320164…000000000000000000000000 (639 digits)2175 \u00d7 369 \u00d7 529 \u00d7 718 \u00d7 1119 \u00d7 138 \u00d7 179 \u00d7 197 \u00d7 239 \u00d7 293 \u00d7 … \u00d7 583367 \u00d7 1609669 \u00d7 3500201 \u00d7 119782433 \u00d7 212601841 \u00d7 2664097031 \u00d7 2931542417 \u00d7 43872038849 \u00d7 374857981681 \u00d7 4534166740403 (115 distinct prime factors)George Woltman, 2013[1]11251850413483992918774837…000000000000000000000000 (1907 digits)2468 \u00d7 3140 \u00d7 566 \u00d7 749 \u00d7 1140 \u00d7 1331 \u00d7 1711 \u00d7 1912 \u00d7 239 \u00d7 297 \u00d7 … \u00d7 25922273669242462300441182317 \u00d7 15428152323948966909689390436420781 \u00d7 420391294797275951862132367930818883361 \u00d7 23735410086474640244277823338130677687887 \u00d7 628683935022908831926019116410056880219316806841500141982334538232031397827230330241 (246 distinct prime factors)George Woltman, 2001[1]Properties[edit]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\u00a0+\u20091)-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.Odd multiply perfect numbers[edit]It is unknown whether there are any odd multiply perfect numbers other than 1. However if an odd k-perfect number n exists where k > 2, then it must satisfy the following conditions:[2]The largest prime factor is \u2265\u2009100129The second largest prime factor is \u2265\u20091009The third largest prime factor is \u2265\u2009101In little-o notation, the number of multiply perfect numbers less than x is o(x\u03b5){displaystyle o(x^{varepsilon })} for all \u03b5 > 0.[2]The number of k-perfect numbers n for n \u2264 x is less than cxc\u2032log\u2061log\u2061log\u2061x\/log\u2061log\u2061x{displaystyle cx^{c’log log log x\/log log x}}, where c and c’ are constants independent of k.[2]Under the assumption of the Riemann hypothesis, the following inequality is true for all k-perfect numbers n, where k > 3"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/multiply-perfect-number-wikipedia\/#breadcrumbitem","name":"Multiply perfect number – Wikipedia"}}]}]