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Highly Totient Number
A highly totient number k is an integer that has more solutions to the equation \phi(x) = k, where \phi is Euler's totient function, than any integer smaller than it. The first few highly totient numbers are 1, 2, 4, 8, 12, 24, 48, 72, 144, 240, 432, 480, 576, 720, 1152, 1440 , with 2, 3, 4, 5, 6, 10, 11, 17, 21, 31, 34, 37, 38, 49, 54, and 72 totient solutions respectively. The sequence of highly totient numbers is a subset of the sequence of smallest number k with exactly n solutions to \phi(x) = k. The totient of a number x, with prime factorization x=\prod_i p_i^, is the product: :\phi(x)=\prod_i (p_i-1)p_i^. Thus, a highly totient number is a number that has more ways of being expressed as a product of this form than does any smaller number. The concept is somewhat analogous to that of highly composite numbers, and in the same way that 1 is the only odd highly composite number, it is also the only odd highly totient number (indeed, the only odd number to not be a n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Totient Function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In other words, it is the number of integers in the range for which the greatest common divisor is equal to 1. The integers of this form are sometimes referred to as totatives of . For example, the totatives of are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since and . Therefore, . As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring \Z/n\Z). It is also used for defining the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
240 (number)
240 (two hundred ndforty) is the natural number following 239 and preceding 241. Mathematics 240 is a pronic number, since it can be expressed as the product of two consecutive integers, 15 and 16. It is a semiperfect number, equal to the concatenation of two of its proper divisors (24 and 40). It is also the 12th highly composite number, with 20 divisors in total, more than any smaller number; and a refactorable number or tau number, since one of its divisors is 20, which divides 240 evenly. 240 is the aliquot sum of only two numbers: 120 and 57121 (or 2392); and is part of the 12161-aliquot tree that goes: 120, 240, 504, 1056, 1968, 3240, 7650, 14112, 32571, 27333, 12161, 1, 0. It is the smallest number that can be expressed as a sum of consecutive primes in three different ways: \begin 240 & = 113 + 127 \\ 240 & = 53 + 59 + 61 + 67 \\ 240 & = 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 \\ \end 240 is highly totient, since it has thirty-one totient answers, more than any p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Factorization
In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a composite number, or it is not, in which case it is a prime number. For example, is a composite number because , but is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example . Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer using mental or pen-and-paper arithmetic, the simplest method is trial division: checking if the number is divisible by prime numbers , , , and so on, up to the square root of . For larger numbers, especially when using a computer, various more sophis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nontotient
In number theory, a nontotient is a positive integer ''n'' which is not a totient number: it is not in the range of Euler's totient function φ, that is, the equation φ(''x'') = ''n'' has no solution ''x''. In other words, ''n'' is a nontotient if there is no integer ''x'' that has exactly ''n'' coprimes below it. All odd numbers are nontotients, except 1, since it has the solutions ''x'' = 1 and ''x'' = 2. The first few even nontotients are this sequence: : 14, 26, 34, 38, 50, 62, 68, 74, 76, 86, 90, 94, 98, 114, 118, 122, 124, 134, 142, 146, 152, 154, 158, 170, 174, 182, 186, 188, 194, 202, 206, 214, 218, 230, 234, 236, 242, 244, 246, 248, 254, 258, 266, 274, 278, 284, 286, 290, 298, ... The least value of ''k'' such that the totient of ''k'' is ''n'' are (0 if no such ''k'' exists) are this sequence: :1, 3, 0, 5, 0, 7, 0, 15, 0, 11, 0, 13, 0, 0, 0, 17, 0, 19, 0, 25, 0, 23, 0, 35, 0, 0, 0, 29, 0, 31, 0, 51, 0, 0, 0, 37, 0, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Highly Composite Number
A highly composite number is a positive integer that has more divisors than all smaller positive integers. If ''d''(''n'') denotes the number of divisors of a positive integer ''n'', then a positive integer ''N'' is highly composite if ''d''(''N'') > ''d''(''n'') for all ''n'' < ''N''. For example, 6 is highly composite because ''d''(6)=4, and for ''n''=1,2,3,4,5, you get ''d''(''n'')=1,2,2,3,2, respectively, which are all less than 4. A related concept is that of a largely composite number, a positive integer that has at least as many divisors as all smaller positive integers. The name can be somewhat misleading, as the first two highly composite numbers (1 and 2) are not actually composite numbers; however, all further terms are. Ramanujan wrote a paper on highly composite numbers in 1915. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Factorization
In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a composite number, or it is not, in which case it is a prime number. For example, is a composite number because , but is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example . Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer using mental or pen-and-paper arithmetic, the simplest method is trial division: checking if the number is divisible by prime numbers , , , and so on, up to the square root of . For larger numbers, especially when using a computer, various more sophis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
720 (number)
720 (seven hundred ndtwenty) is the natural number following 719 and preceding 721. In mathematics 720 is: *6! ( 6 factorial). *a composite number with 30 divisors, more than any number below, making it the 14th highly composite number. *a highly abundant number. *a Harshad number in every base from binary to decimal. *the smallest number to be palindromic in 16 bases. *a 241- gonal number. 720 is expressible as the product of consecutive integers in two different ways: and . There are 49 solutions to the equation , more than any integer below it, making 720 a highly totient number. In other fields 720 is: * A common vertical display resolution for HDTV (see 720p). * 720° is two full rotations; the term "720" refers to a skateboarding trick. * 720° is also the name of a skateboarding video game. * 720 is a dual area code in the Denver Metro Area along with 303. * 720° is the sum of all the defects of any polyhedron In geometry, a polyhedron (: polyhedra or polyhedr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
144 (number)
144 (one hundred [and] forty-four) is the natural number following 143 (number), 143 and preceding 145 (number), 145. It is coincidentally both the Square number, square of twelve (a dozen dozens, or one Gross (unit), gross) and the twelfth Fibonacci number, and the only Triviality (mathematics), nontrivial number in the sequence that is square. Mathematics 144 is a highly totient number. 144 is the smallest number whose fifth Power (mathematics), power is a sum of four (smaller) fifth powers. This solution was found in 1966 by L. J. Lander and T. R. Parkin, and disproved Euler's sum of powers conjecture. It was famously published in a paper by both authors, whose body consisted of only two sentences: It is also a square (12·12=144). In other fields * 1:144 scale is a scale used for some scale models. * Several computers use 144Hz. References * Wells, D. ''The Penguin Dictionary of Curious and Interesting Numbers''. London: Penguin Group. (1987): 139–140. External link ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1 (number)
1 (one, unit, unity) is a number, numeral, and glyph. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property has led to its unique uses in other fields, ranging from science to sports, where it commonly denotes the first, leading, or top thing in a group. 1 is the unit of counting or measurement, a determiner for singular nouns, and a gender-neutral pronoun. Historically, the representation of 1 evolved from ancient Sumerian and Babylonian symbols to the modern Arabic numeral. In mathematics, 1 is the multiplicative identity, meaning that any number multiplied by 1 equals the same number. 1 is by convention not considered a prime number. In digital technology, 1 represents the "on" state in binary code, the foundation of computing. Philosophically, 1 symbolizes the ultimate reality or source of existence in various traditions. In mathematics The number 1 is the first natural number after 0. Each natural ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
72 (number)
72 (seventy-two) is the natural number following 71 and preceding 73. It is half a gross or six dozen (i.e., 60 in duodecimal). In mathematics Seventy-two is a pronic number, as it is the product of 8 and 9. It is the smallest Achilles number, as it is a powerful number that is not itself a power. 72 is an abundant number. With exactly twelve positive divisors, including 12 (one of only two sublime numbers), 72 is also the twelfth member in the sequence of refactorable numbers. As no smaller number has more than 12 divisors, 72 is a largely composite number. 72 has an Euler totient of 24. It is a highly totient number, as there are 17 solutions to the equation φ(''x'') = 72, more than any integer under 72. It is equal to the sum of its preceding smaller highly totient numbers 24 and 48, and contains the first six highly totient numbers 1, 2, 4, 8, 12 and 24 as a subset of its proper divisors. 144, or twice 72, is also highly totient, as is 576, the square ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
48 (number)
48 (forty-eight) is the natural number following 47 and preceding 49. It is one third of a gross, or four dozens. In mathematics 48 is a highly composite number, and a Størmer number. By a classical result of Honsberger, the number of incongruent integer-sided triangles of perimeter m is given by the equations \tfrac for even m, and \tfrac for odd m. 48 is the order of full octahedral symmetry, which describes three-dimensional mirror symmetries associated with the regular octahedron and cube. There are 48 symmetries of a cube. In other fields Forty-eight may also refer to: * In Chinese numerology, 48 is an auspicious number meaning 'determined to prosper', or simply 'prosperity', which is good for business. * '48 is a slang term in Palestinian Arabic for parts of Israel or Palestine not under the control of the State of Palestine Palestine, officially the State of Palestine, is a country in West Asia. Recognized by International recognition of Pal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
