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Refactorable Number
A refactorable number or tau number is an integer ''n'' that is divisible by the count of its divisors, or to put it algebraically, ''n'' is such that \tau(n)\mid n. The first few refactorable numbers are listed in as : 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225, 228, 232, 240, 248, 252, 276, 288, 296, ... For example, 18 has 6 divisors (1 and 18, 2 and 9, 3 and 6) and is divisible by 6. There are infinitely many refactorable numbers. Properties Cooper and Kennedy proved that refactorable numbers have natural density zero. Zelinsky proved that no three consecutive integers can all be refactorable. Colton proved that no refactorable number is perfect. The equation \gcd(n,x) = \tau(n) has solutions only if n is a refactorable number, where \gcd is the greatest common divisor function. Let T(x) be the number of refactorable numbers which are at most x. The problem of determining an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
96 (number)
96 (ninety-six) is the natural number following 95 and preceding 97. It is a number that appears the same when turned upside down. In mathematics 96 is: * an octagonal number. * a refactorable number. * an untouchable number. * a semiperfect number since it is a multiple of 6. * an abundant number since the sum of its proper divisors is greater than 96. * the fourth Granville number and the second non-perfect Granville number. The next Granville number is 126, the previous being 24. * the sum of Euler's totient function φ(''x'') over the first seventeen integers. * strobogrammatic in bases 10 (9610), 11 (8811) and 95 (1195). * palindromic in bases 11 (8811), 15 (6615), 23 (4423), 31 (3331), 47 (2247) and 95 (1195). * an Erdős–Woods number, since it is possible to find sequences of 96 consecutive integers such that each inner member shares a factor with either the first or the last member. * divisible by the number of prime numbers (24) below 96. Every integer greater ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
232 (number)
232 (two hundred ndthirty-two) is the natural number following 231 and preceding 233. In mathematics 232 is both a central polygonal number and a cake number. It is both a decagonal number and a centered 11-gonal number. It is also a refactorable number, a Motzkin sum, an idoneal number, a Riordan number and a noncototient. 232 is a telephone number: in a system of seven telephone users, there are 232 different ways of pairing up some of the users. There are also exactly 232 different eight-vertex connected indifference graphs, and 232 bracelets A bracelet is an article of jewellery that is worn around the wrist. Bracelets may serve different uses, such as being worn as an ornament. When worn as ornaments, bracelets may have a supportive function to hold other items of decoration, suc ... with eight beads of one color and seven of another. Because this number has the form , it follows that there are exactly 232 different functions from a set of four elements to a prope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
228 (number)
228 (two hundred ndtwenty-eight) is the natural number following 227 and preceding 229. In mathematics 228 is a refactorable number and a practical number. There are 228 matchings in a ladder graph with five rungs. 228 is the smallest even number ''n'' such that the numerator of the ''n''th Bernoulli number is divisible by a nontrivial square number that is relatively prime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ... to ''n''. The binary form of 228 contains all the two digit binary numbers in sequence from highest to lowest (11 10 01 00). References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
225 (number)
225 (two hundred ndtwenty-five) is the natural number following 224 and preceding 226. In mathematics 225 is the smallest number that is a polygonal number in five different ways. It is a square number , an octagonal number, and a squared triangular number . As the square of a double factorial, counts the number of permutations of six items in which all cycles have even length, or the number of permutations in which all cycles have odd length. And as one of the Stirling numbers of the first kind, it counts the number of permutations of six items with exactly three cycles. 225 is a highly composite odd number, meaning that it has more divisors than any smaller odd numbers. After 1 and 9, 225 is the third smallest number ''n'' for which , where ''σ'' is the sum of divisors function and ''φ'' is Euler's totient function. 225 is a refactorable number. 225 is the smallest square number to have one of every digit in some number base (225 is 3201 in base 4) 225 is the fir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
204 (number)
204 (two hundred ndfour) is the natural number following 203 and preceding 205. In mathematics 204 is a refactorable number. 204 is a square pyramidal number: 204 balls may be stacked in a pyramid whose base is an 8 × 8 square. Its square, 2042 = 41616, is the fourth square triangular number. As a figurate number, 204 is also a nonagonal number and a truncated triangular pyramid number. 204 is a member of the Mian-Chowla sequence. There are exactly 204 irreducible quintic polynomials over a four-element field, exactly 204 ways to place three non-attacking chess queens on a 5 × 5 board, exactly 204 squares of an infinite chess move that are eight knight's moves from the center, exactly 204 strings of length 11 over a three-letter alphabet with no consecutively-repeated substring, and exactly 204 ways of immersing an oriented circle into the oriented plane so that it has four double points. Both 204 and its square are sums of a pair of twin p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
184 (number)
184 (one hundred ndeighty-four) is the natural number following 183 and preceding 185. In mathematics There are 184 different Eulerian graphs on eight unlabeled vertices, and 184 paths by which a chess rook can travel from one corner of a 4 × 4 chessboard to the opposite corner without passing through the same square twice. 184 is also a refactorable number. In other fields Some physicists have proposed that 184 is a magic number for neutrons in atomic nuclei. In poker, with one or more jokers as wild cards, there are 184 different straight flushes. See also * The year AD 184 or 184 BC * List of highways numbered 184 The following highways are numbered 184: Ireland * R184 road (Ireland) Japan * Japan National Route 184 Poland * Voivodeship road 184 (Poland), Voivodeship road 184 United States * Interstate 184 * Alabama State Route 184 * Arkansas Highway ... * References {{DEFAULTSORT:184 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
180 (number)
180 (one hundred ndeighty) is the natural number following 179 and preceding 181. In mathematics 180 is an abundant number, with its proper divisors summing up to 366. 180 is also a highly composite number, a positive integer with more divisors than any smaller positive integer. One of the consequences of 180 having so many divisors is that it is a practical number, meaning that any positive number smaller than 180 that is not a divisor of 180 can be expressed as the sum of some of 180's divisors. 180 is a Harshad number and a refactorable number. 180 is the sum of two square numbers: 122 + 62. It can be expressed as either the sum of six consecutive prime numbers: 19 + 23 + 29 + 31 + 37 + 41, or the sum of eight consecutive prime numbers: 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37. 180 is an Ulam number, which can be expressed as a sum of earlier terms in the Ulam sequence only as 177 + 3. 180 is a 61- gonal number. Half a circle has 180 degrees, and thus a U-turn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
156 (number)
156 (one hundred ndfifty-six) is the natural number, following 155 and preceding 157. In mathematics 156 is an abundant number, a pronic number, a dodecagonal number, and a refactorable number. 156 is the number of graphs on 6 unlabeled nodes. 156 is a repdigit in base 5 (1111), and also in bases 25, 38, 51, 77, and 155. 156 degrees is the internal angle of a pentadecagon. In the military * Convoy HX-156 was the 156th of the numbered series of World War II HX convoys of merchant ships from Halifax, Nova Scotia to Liverpool during World War II * The Fieseler Fi 156 Storch was a small German liaison aircraft during World War II * The * was a United States Navy T2 tanker during World War II * was a United States Navy cargo ship during World War II * was a United States Navy during World War II * was a United States Navy ship during World War II * was a United States Navy during World War II * was a United States Navy during World War II * was a United States Navy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
152 (number)
152 (one hundred ndfifty-two) is the natural number following 151 and preceding 153. In mathematics 152 is the sum of four consecutive primes (31 + 37 + 41 + 43). It is a nontotient since there is no integer with 152 coprimes below it. 152 is a refactorable number since it is divisible by the total number of divisors it has, and in base 10 it is divisible by the sum of its digits, making it a Harshad number. Recently, the smallest repunit probable prime in base 152 was found, it has 589570 digits. The number of surface points on a 6*6*6 cube is 152. In the military * Focke-Wulf Ta 152 was a Luftwaffe high-altitude interceptor fighter aircraft during World War II * was a United States Navy during World War II * was a United States Navy during World War II * was a United States Navy supply ship during World War II * was a United States Navy during World War II * was a United States Navy ship during World War II * was a United States Navy during World War II * wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
136 (number)
136 (one hundred ndthirty six) is the natural number following 135 and preceding 137. In mathematics 136 is itself a factor of the Eddington number. With a total of 8 divisors, 8 among them, 136 is a refactorable number. It is a composite number. 136 is a centered triangular number and a centered nonagonal number. The sum of the ninth row of Lozanić's triangle is 136. 136 is a self-descriptive number in base 4, and a repdigit in base 16. In base 10, the sum of the cubes of its digits is 1^3 + 3^3 + 6^3 = 244. The sum of the cubes of the digits of 244 is 2^3 + 4^3 + 4^3 = 136. 136 is a triangular number, because it's the sum of the first 16 positive integers. In the military * Force 136 branch of the British organization, the Special Operations Executive (SOE), in the South-East Asian Theatre of World War II * USNS ''Mission Soledad'' (T-AO-136) was a United States Navy ''Mission Buenaventura''-class fleet oiler during World War II * USS ''Admirable'' (AM-136) was a United ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
132 (number)
132 (one hundred ndthirty-two) is the natural number following 131 and preceding 133. In mathematics 132 is the sixth Catalan number. It is a pronic number, the product of 11 and 12. As it has 12 divisors total, 132 is a refactorable number. If you take the sum of all 2-digit numbers you can make from 132, you get 132: 12 + 13 + 21 + 23 + 31 + 32 = 132. 132 is the smallest number with this property, which is shared by 264, 396 and 35964 (see digit-reassembly number). The exceptional outer automorphism of symmetric group ''S''6 uniquely maps vertices to factorizations and edges to partitions in the graph factors of the complete graph ''K''6, which yields 132 blocks in Steiner system S(5,6,12). In other fields 132 is also: * The year AD 132 or 132 BC * 132 AH is a year in the Islamic calendar that corresponds to 749 – 750 CE * OGLE-TR-132 is a magnitude 15.72 star in the star fields of the constellation Carina * 132 Aethra is a M-type main belt asteroid * Sonnet 132 by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
