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Primorial
In mathematics, and more particularly in number theory, primorial, denoted by "", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubner, draws an analogy to ''primes'' similar to the way the name "factorial" relates to ''factors''. Definition for prime numbers For the th prime number , the primorial is defined as the product of the first primes: :p_n\# = \prod_^n p_k, where is the th prime number. For instance, signifies the product of the first 5 primes: :p_5\# = 2 \times 3 \times 5 \times 7 \times 11= 2310. The first few primorials are: : 1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690... . Asymptotically, primorials grow according to: :p_n\# = e^, where is Little O notation. Definition for natural numbers In general, for a positive integer , its primorial, , is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primorial N Plot
In mathematics, and more particularly in number theory, primorial, denoted by "", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubner, draws an analogy to ''primes'' similar to the way the name "factorial" relates to ''factors''. Definition for prime numbers For the th prime number , the primorial is defined as the product of the first primes: :p_n\# = \prod_^n p_k, where is the th prime number. For instance, signifies the product of the first 5 primes: :p_5\# = 2 \times 3 \times 5 \times 7 \times 11= 2310. The first few primorials are: : 1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690... . Asymptotically, primorials grow according to: :p_n\# = e^, where is Little O notation. Definition for natural numbers In general, for a positive integer , its primorial, , is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primorial Pn Plot
In mathematics, and more particularly in number theory, primorial, denoted by "", is a Function (mathematics), function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubner, draws an analogy to ''primes'' similar to the way the name "factorial" relates to ''factors''. Definition for prime numbers For the th prime number , the primorial is defined as the product of the first primes: :p_n\# = \prod_^n p_k, where is the th prime number. For instance, signifies the product of the first 5 primes: :p_5\# = 2 \times 3 \times 5 \times 7 \times 11= 2310. The first few primorials are: :1 (number), 1, 2 (number), 2, 6 (number), 6, 30 (number), 30, 210 (number), 210, 2310 (number), 2310, 30030, 510510, 9699690... . Asymptotically, primorials grow according to: :p_n\# = e^, where is Little O notation. ... [...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 Factor
A prime number (or a prime) is a natural number greater than 1 that is not a 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 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 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, and the AKS primality test, which always pro ... [...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] |
2310 (number)
2000 (two thousand) is a natural number following 1999 and preceding 2001. It is: :*the highest number expressible using only two unmodified characters in Roman numerals (MM) :*an Achilles number :*smallest four digit eban number :*the sum of all the nban numbers in the sequence Selected numbers in the range 2001–2999 2001 to 2099 * 2001 – sphenic number * 2002 – palindromic number in decimal, base 76, 90, 142, and 11 other non-trivial bases * 2003 – Sophie Germain prime and the smallest prime number in the 2000s * 2004 – Area of the 24tcrystagon* 2005 – A vertically symmetric number * 2006 – number of subsets of with relatively prime elements * 2007 – 22007 + 20072 is prime * 2008 – number of 4 × 4 matrices with nonnegative integer entries and row and column sums equal to 3 * 2009 = 74 − 73 − 72 * 2010 – number of compositions of 12 into relatively prime parts * 2011 – sexy prime with 2017, sum of eleven consecutive primes: 2011 = 157 + 163 + 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Function
In mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function or is given by :\vartheta(x) = \sum_ \log p where \log denotes the natural logarithm, with the sum extending over all prime numbers that are less than or equal to . The second Chebyshev function is defined similarly, with the sum extending over all prime powers not exceeding :\psi(x) = \sum_\sum_\log p = \sum_ \Lambda(n) = \sum_\left\lfloor\log_p x\right\rfloor\log p, where is the von Mangoldt function. The Chebyshev functions, especially the second one , are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, (see the exact formula below.) Both Chebyshev functions are asymptotic to , a statement equivalent to the prime number theorem. Tchebycheff function, Chebyshev utility function, or weighted Tchebycheff scalariz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial
In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \times (n-3) \times \cdots \times 3 \times 2 \times 1 \\ &= n\times(n-1)!\\ \end For example, 5! = 5\times 4! = 5 \times 4 \times 3 \times 2 \times 1 = 120. The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book ''Sefer Yetzirah''. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of n distinct objects: there In mathematical analysis, factorials are used in power series for the ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primes In Arithmetic Progression
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by a_n = 3 + 4n for 0 \le n \le 2. According to the Green–Tao theorem, there exist arbitrarily long arithmetic progressions in the sequence of primes. Sometimes the phrase may also be used about primes which belong to an arithmetic progression which also contains composite numbers. For example, it can be used about primes in an arithmetic progression of the form an + b, where ''a'' and ''b'' are coprime which according to Dirichlet's theorem on arithmetic progressions contains infinitely many primes, along with infinitely many composites. For any integer k\geq 3, an AP-''k'' (also called PAP-''k'') is any sequence of k primes in arithmetic progression. An AP-k can be written as k primes of the form an+b, for fixed integers a (called the common difference) and b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
210 (number)
210 (two hundred ndten) is the natural number following 209 and preceding 211. Mathematics 210 is an abundant number, and Harshad number. It is the product of the first four prime numbers ( 2, 3, 5, and 7), and thus a primorial, where it is the least common multiple of these four prime numbers. 210 is the first primorial number greater than 2 which is not adjacent to 2 primes (211 is prime, but 209 is not). It is the sum of eight consecutive prime numbers, between 13 and the thirteenth prime number: Wells, D. (1987). ''The Penguin Dictionary of Curious and Interesting Numbers'' (p. 143). London: Penguin Group. It is the 20th triangular number (following 190 and preceding 231), a pentagonal number (following 176 and preceding 247), and the second smallest to be both triangular and pentagonal (the first is 1; the third is 40755). It is also an idoneal number, a pentatope number, a pronic number, and an untouchable number. 210 is also the third 71-gonal number, pre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
30 (number)
30 (thirty) is the natural number following 29 and preceding 31. In mathematics 30 is an even, composite, and pronic number. With 2, 3, and 5 as its prime factors, it is a regular number and the first sphenic number, the smallest of the form , where is a prime greater than 3. It has an aliquot sum of 42; within an aliquot sequence of thirteen composite numbers (30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0) to the Prime in the 3-aliquot tree. From 1 to the number 30, this is the longest Aliquot Sequence. It is also: * A semiperfect number, since adding some subsets of its divisors (e.g., 5, 10 and 15) equals 30. * A primorial. * A Harshad number in decimal. * Divisible by the number of prime numbers ( 10) below it. * The largest number such that all coprimes smaller than itself, except for 1, are prime. * The sum of the first four squares, making it a square pyramidal number. * The number of vertices in the Tutte–Coxeter graph. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Engel Expansion
The Engel expansion of a positive real number ''x'' is the unique non-decreasing sequence of positive integers (a_1,a_2,a_3,\dots) such that :x=\frac+\frac+\frac+\cdots = \frac\!\left(1 + \frac\!\left(1 + \frac\left(1+\cdots\right)\right)\right) For instance, Euler's number ''e'' has the Engel expansion :1, 1, 2, 3, 4, 5, 6, 7, 8, ... corresponding to the infinite series :e=\frac+\frac+\frac+\frac+\frac+\cdots Rational numbers have a finite Engel expansion, while irrational numbers have an infinite Engel expansion. If ''x'' is rational, its Engel expansion provides a representation of ''x'' as an Egyptian fraction. Engel expansions are named after Friedrich Engel, who studied them in 1913. An expansion analogous to an Engel expansion, in which alternating terms are negative, is called a Pierce expansion. Engel expansions, continued fractions, and Fibonacci observe that an Engel expansion can also be written as an ascending variant of a continued fraction: :x = \cfra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
