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Triperfect 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 (or 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 . A number that is for a certain ''k'' is called a multiply perfect number. As of 2014, 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, ... . 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 number. Smallest known ''k''-perfect numbers The following table gives an overv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiply Perfect Number Cuisenaire Rods 6
Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a '' product''. Multiplication is often denoted by the cross symbol, , by the mid-line dot operator, , by juxtaposition, or, in programming languages, by an asterisk, . The multiplication of whole numbers may be thought of as repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the ''multiplicand'', as the quantity of the other one, the ''multiplier''; both numbers can be referred to as ''factors''. This is to be distinguished from ''terms'', which are added. :a\times b = \underbrace_ . Whether the first factor is the multiplier or the multiplicand may be ambiguous or depend upon context. For example, the expression 3 \times 4 , can be phrased as "3 times 4" and evaluated as 4+4+4, where 3 is the multiplier, but a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descartes Number
In number theory, a Descartes number is an odd number which would have been an odd perfect number if one of its composite factors were prime. They are named after René Descartes who observed that the number would be an odd perfect number if only were a prime number, since the sum-of-divisors function for would satisfy, if 22021 were prime, :\begin \sigma(D) &= (3^2+3+1)\cdot(7^2+7+1)\cdot(11^2+11+1)\cdot(13^2+13+1)\cdot(22021+1) \\ &= (13)\cdot(3\cdot19)\cdot(7\cdot19)\cdot(3\cdot61)\cdot(22\cdot1001) \\ &= 3^2\cdot7\cdot13\cdot19^2\cdot61\cdot(22\cdot7\cdot11\cdot13) \\ &= 2 \cdot (3^2\cdot7^2\cdot11^2\cdot13^2) \cdot (19^2\cdot61) \\ &= 2 \cdot (3^2\cdot7^2\cdot11^2\cdot13^2) \cdot 22021 = 2D, \end where we ignore the fact that 22021 is composite (). A Descartes number is defined as an odd number where and are coprime and , whence is taken as a 'spoof' prime. The example given is the only one currently known. If is an odd almost perfect number, that is, and is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramaswamy S
Ramaswamy S is an Indian-American structural biologist of Indian origin. He is currently professor of biological sciences, professor at the Weldon School of Biomedical Engineering and director of the Bindley Bioscience Center at Purdue University. He was a senior professor at Institute of stem cell biology and regenerative medicine (inStem) from 2009 to 2019. Ramaswamy S also abbreviated as "Rams" by his students and colleagues, held the founding dean's position at inStem from May 2009. He was also the founding CEO of Center for Cellular and molecular platforms (C-CAMP), Bangalore-Biocluster from May 2009 to May 2016. His major role at National Center for Biological Sciences (NCBS) was to bridge the gap between research and application. He works in close coordination with Department of Biotechnology (DBT), and other ministries of the Government of India. Rams obtained his Ph.D. in molecular biophysics under the guidance of Prof. M. R. N. Murthy from the Indian Institute of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime Integers
In number theory, 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 equivalent to their greatest common divisor (GCD) being 1. One says also ''is prime to'' or ''is coprime with'' . The numbers 8 and 9 are coprime, despite the fact that neither—considered individually—is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing When the integers and are coprime, the standard way of expressing this fact in mathematical notation is to indicate that their greatest common divisor is one, by the formula or . In their 1989 textbook ''Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed an alternative ... [...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] |
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] |
Square Number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of Figurate number, figurate numbers (other examples being Cube (algebra), cube numbers and triangular numbers). In the Real number, real number system, square numbers are non-negative. A non-negative integer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Problem
In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is known). In the history of science, some of these supposed open problems were "solved" by means of showing that they were not well-defined. In mathematics, many open problems are concerned with the question of whether a certain definition is or is not consistent. Two notable examples in mathematics that have been solved and ''closed'' by researchers in the late twentieth century are Fermat's Last Theorem and the four-color theorem.K. Appel and W. Haken (1977), "Every planar map is four colorable. Part I. Discharging", ''Illinois J. Math'' 21: 429–490. K. Appel, W. Haken, and J. Koch (1977), "Every planar map is four colorable. Part II. Reducibility", ''Illinois J. Math'' 21: 491–567. An important open mathematics problem solved ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Omega Function
In number theory, the prime omega functions \omega(n) and \Omega(n) count the number of prime factors of a natural number n. The number of ''distinct'' prime factors is assigned to \omega(n) (little omega), while \Omega(n) (big omega) counts the ''total'' number of prime factors with multiplicity (see arithmetic function). That is, if we have a prime factorization of n of the form n = p_1^ p_2^ \cdots p_k^ for distinct primes p_i (1 \leq i \leq k), then the prime omega functions are given by \omega(n) = k and \Omega(n) = \alpha_1 + \alpha_2 + \cdots + \alpha_k. These prime-factor-counting functions have many important number theoretic relations. Properties and relations The function \omega(n) is additive and \Omega(n) is completely additive. Little omega has the formula \omega(n)=\sum_ 1, where notation indicates that the sum is taken over all primes that divide , without multiplicity. For example, \omega(12)=\omega(2^2 3)=2. Big omega has the formulas \Omega(n) =\sum_ 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Of Divisors
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum. A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function. Definition The sum of positive divisors function ''σ''''z''(''n''), for a real or complex number ''z'', is defined as the sum of the ''z''th powers of the positive divisors of ''n''. It can be expressed in sigma notation as :\sigma_z(n)=\sum_ d^z\,\! , where is shorthand for "''d'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler–Mascheroni Constant
Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by : \begin \gamma &= \lim_\left(-\log n + \sum_^n \frac1\right)\\ px&=\int_1^\infty\left(-\frac1x+\frac1\right)\,\mathrm dx. \end Here, represents the floor function. The numerical value of Euler's constant, to 50 decimal places, is: History The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled ''De Progressionibus harmonicis observationes'' (Eneström Index 43), where he described it as "worthy of serious consideration". Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Euler used the notations and for the constant. The Italian mathematician Lorenzo Mascheroni attempted to calculate the constant to 32 dec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
