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Unitary Perfect Number
A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself. (A divisor ''d'' of a number ''n'' is a unitary divisor if ''d'' and ''n''/''d'' share no common factors). The number 6 is the only number that is both a perfect number and a unitary perfect number. Known examples The number 60 is a unitary perfect number because 1, 3, 4, 5, 12, 15, and 20 are its proper unitary divisors, and 1 + 3 + 4 + 5 + 12 + 15 + 20 = 60. The first five, and only known, unitary perfect numbers are: * 6 = 2 \times 3 * 60 = 2^2 \times 3 \times 5 * 90 = 2 \times 3^2 \times 5 * 87360 = 2^6 \times 3 \times 5 \times 7 \times 13, and *146361946186458562560000 = 2^ \times 3 \times 5^4 \times 7 \times 11 \times 13 \times 19 \times 37 \times 79 \times 109 \times 157 \times 313 . The respective sums of their proper unitary divisors are as follows: * 6 = 1 + 2 + 3 * 60 = 1 + 3 + 4 + 5 + 12 + 15 + 20 * 90 = 1 + 2 + 5 + 9 + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proper Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Definition An integer n is divisible by a nonzero integer m if there exists an integer k such that n=km. This is written as : m\mid n. This may be read as that m divides n, m is a divisor of n, m is a factor of n, or n is a multiple of m. If m does not divide n, then the notation is m\not\mid n. There are two conventions, distinguished by whether m is permitted to be zero: * With the convention without an additional constraint on m, m \mid 0 for every integer m. * With the convention that m be nonzero, m \mid 0 for every nonzero integer m. General Divisors can be negative as well as positive, although often the term is restricted to posi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Divisor
In mathematics, a natural number ''a'' is a unitary divisor (or Hall divisor) of a number ''b'' if ''a'' is a divisor of ''b'' and if ''a'' and \frac are coprime, having no common factor other than 1. Equivalently, a divisor ''a'' of ''b'' is a unitary divisor if and only if every prime factor of ''a'' has the same multiplicity in ''a'' as it has in ''b''. The concept of a unitary divisor originates from R. Vaidyanathaswamy (1931), who used the term block divisor. Example The integer 5 is a unitary divisor of 60, because 5 and \frac=12 have only 1 as a common factor. On the contrary, 6 is a divisor but not a unitary divisor of 60, as 6 and \frac=10 have a common factor other than 1, namely 2. Sum of unitary divisors The sum-of-unitary-divisors function is denoted by the lowercase Greek letter sigma thus: σ*(''n''). The sum of the ''k''-th powers of the unitary divisors is denoted by σ*''k''(''n''): :\sigma_k^*(n) = \sum_ \!\! d^k. It is a multiplicative function. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Definition An integer n is divisible by a nonzero integer m if there exists an integer k such that n=km. This is written as : m\mid n. This may be read as that m divides n, m is a divisor of n, m is a factor of n, or n is a multiple of m. If m does not divide n, then the notation is m\not\mid n. There are two conventions, distinguished by whether m is permitted to be zero: * With the convention without an additional constraint on m, m \mid 0 for every integer m. * With the convention that m be nonzero, m \mid 0 for every nonzero integer m. General Divisors can be negative as well as positive, although often the term is restricted to posi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, since 1 + 2 + 4 + 7 + 14 = 28. The first four perfect numbers are 6 (number), 6, 28 (number), 28, 496 (number), 496 and 8128 (number), 8128. The sum of proper divisors of a number is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors; in symbols, \sigma_1(n)=2n where \sigma_1 is the sum-of-divisors function. This definition is ancient, appearing as early as Euclid's Elements, Euclid's ''Elements'' (VII.22) where it is called (''perfect'', ''ideal'', or ''complete number''). Euclid also proved a formation rule (IX.36) whereby \frac is an even perfect number whenever q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
60 (number)
60 (sixty) () is the natural number following 59 and preceding 61. Being three times 20, it is called '' threescore'' in older literature ('' kopa'' in Slavic, ''Schock'' in Germanic). In mathematics 60 is the 4th superior highly composite number, the 4th colossally abundant number, the 9th highly composite number, a unitary perfect number, and an abundant number. It is the smallest number divisible by the numbers 1 to 6. The smallest group that is not solvable is the alternating group A5, which has 60 elements. There are 60 one-sided hexominoes, the polyominoes made from six squares. There are 60 seconds in a minute, as well as 60 minutes in a degree. In science and technology The first fullerene to be discovered was buckminsterfullerene C60, an allotrope of carbon with 60 atoms in each molecule, arranged in a truncated icosahedron. This ball is known as a buckyball, and looks like a soccer ball. The atomic number of neodymium is 60, and cobalt-60 (60Co) is a radio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6 (number)
6 (six) is the natural number following 5 and preceding 7. It is a composite number and the smallest perfect number. In mathematics A six-sided polygon is a hexagon, one of the three regular polygons capable of tiling the plane. A hexagon also has 6 edges as well as 6 internal and external angles. 6 is the second smallest composite number. It is also the first number that is the sum of its proper divisors, making it the smallest perfect number. It is also the only perfect number that doesn't have a digital root of 1. 6 is the first unitary perfect number, since it is the sum of its positive proper unitary divisors, without including itself. Only five such numbers are known to exist. 6 is the largest of the four all-Harshad numbers. 6 is the 2nd superior highly composite number, the 2nd colossally abundant number, the 3rd triangular number, the 4th highly composite number, a pronic number, a congruent number, a harmonic divisor number, and a semiprime. 6 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
90 (number)
90 (ninety) is the natural number following 89 and preceding 91. In the English language, the numbers 90 and 19 are often confused, as they sound very similar. When carefully enunciated, they differ in which syllable is stressed: 19 /naɪnˈtiːn/ vs 90 /ˈnaɪnti/. However, in dates such as 1999, and when contrasting numbers in the teens and when counting, such as 17, 18, 19, the stress shifts to the first syllable: 19 /ˈnaɪntiːn/. In mathematics Ninety is a pronic number as it is the product of 9 and 10, and along with 12 and 56, one of only a few pronic numbers whose digits in decimal are also successive. 90 is divisible by the sum of its base-ten digits, which makes it the thirty-second Harshad number. Properties of the number *90 is the only number to have an aliquot sum of 144 = 122. *Only three numbers have a set of divisors that generate a sum equal to 90, they are 40, 58, and 89. *90 is also the twentieth abundant and highly abundan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity (mathematics)
In mathematics, parity is the Property (mathematics), property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers with decimals or fractions like 1/2 or 4.6978. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as ... [...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] |
Multiplicative Function
In number theory, a multiplicative function is an arithmetic function f of a positive integer n with the property that f(1)=1 and f(ab) = f(a)f(b) whenever a and b are coprime. An arithmetic function is said to be completely multiplicative (or totally multiplicative) if f(1)=1 and f(ab) = f(a)f(b) holds ''for all'' positive integers a and b, even when they are not coprime. Examples Some multiplicative functions are defined to make formulas easier to write: * 1(n): the constant function defined by 1(n)=1 * \operatorname(n): the identity function, defined by \operatorname(n)=n * \operatorname_k(n): the power functions, defined by \operatorname_k(n)=n^k for any complex number k. As special cases we have ** \operatorname_0(n)=1(n), and ** \operatorname_1(n)=\operatorname(n). * \varepsilon(n): the function defined by \varepsilon(n)=1 if n=1 and 0 otherwise; this is the unit function, so called because it is the multiplicative identity for Dirichlet convolution. Sometimes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Power
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, … . The prime powers are those positive integers that are divisible by exactly one prime number; in particular, the number 1 is not a prime power. Prime powers are also called primary numbers, as in the primary decomposition. Properties Algebraic properties Prime powers are powers of prime numbers. Every prime power (except powers of 2 greater than 4) has a primitive root; thus the multiplicative group of integers modulo ''p''''n'' (that is, the group of units of the ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
