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253 (number)
253 (two hundred ndfifty-three) is the natural number following 252 and preceding 254. In mathematics 253 is: *a semiprime since it is the product of 2 primes. *a triangular number. *a star number. *a centered heptagonal number. *a centered nonagonal number. *a Blum integer In mathematics, a natural number ''n'' is a Blum integer if is a semiprime for which ''p'' and ''q'' are distinct prime numbers congruent to 3 mod 4.Joe Hurd, Blum Integers (1997), retrieved 17 Jan, 2011 from http://www.gilith.com/research/talks/ .... *a member of the 13-aliquot tree. References {{Integers, 2 Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal numbers'', and numbers used for ordering are called '' ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports jersey numbers). Some definitions, including the standard ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural numbers form a set. Many other number sets are built by succ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
252 (number)
252 (two hundred ndfifty-two) is the natural number following 251 and preceding 253. In mathematics 252 is: *the central binomial coefficient \tbinom, the largest one divisible by all coefficients in the previous line *\tau(3), where \tau is the Ramanujan tau function. *\sigma_3(6), where \sigma_3 is the function that sums the cubes of the divisors of its argument: :1^3+2^3+3^3+6^3=(1^3+2^3)(1^3+3^3)=252. *a practical number, *a refactorable number, *a hexagonal pyramidal number. *a member of the Mian-Chowla sequence. There are 252 points on the surface of a cuboctahedron of radius five in the face-centered cubic In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties o ... lattice, 252 ways of writing the number 4 as a sum of six squares of integers, 252 ways of choosing four squares fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
254 (number)
254 (two hundred ndfifty-four) is the natural number following 253 and preceding 255. In mathematics * It is a deficient number, since the sum of its divisors (excluding the same number) is 130 < 254. * It is a . Moreover, in , its name has a semiprime number of syllables. * It is a . * It is a . * It is a [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiprime
In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime numbers, there are also infinitely many semiprimes. Semiprimes are also called biprimes. Examples and variations The semiprimes less than 100 are: Semiprimes that are not square numbers are called discrete, distinct, or squarefree semiprimes: The semiprimes are the case k=2 of the k- almost primes, numbers with exactly k prime factors. However some sources use "semiprime" to refer to a larger set of numbers, the numbers with at most two prime factors (including unit (1), primes, and semiprimes). These are: Formula for number of semiprimes A semiprime counting formula was discovered by E. Noel and G. Panos in 2005. Let \pi_2(n) denote the number of semiprimes less than or equal to n. Then \pi_2(n) = \sum_^ pi(n/p_k) - k + 1 /math> w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in the triangular arrangement with dots on each side, and is equal to the sum of the natural numbers from 1 to . The sequence of triangular numbers, starting with the 0th triangular number, is (This sequence is included in the On-Line Encyclopedia of Integer Sequences .) Formula The triangular numbers are given by the following explicit formulas: T_n= \sum_^n k = 1+2+3+ \dotsb +n = \frac = , where \textstyle is a binomial coefficient. It represents the number of distinct pairs that can be selected from objects, and it is read aloud as " plus one choose two". The first equation can be illustrated using a visual proof. For every triangular number T_n, imagine a "half-square" arrangement of objects corresponding to the triangular n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Number
A star number is a centered figurate number, a centered hexagram (six-pointed star), such as the Star of David, or the board Chinese checkers is played on. The ''n''th star number is given by the formula ''Sn'' = 6''n''(''n'' − 1) + 1. The first 43 star numbers are 1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, 1093, 1261, 1441, 1633, 1837, 2053, 2281, 2521, 2773, 3037, 3313, 3601, 3901, 4213, 4537, 4873, 5221, 5581, 5953, 6337, 6733, 7141, 7561, 7993, 8437, 8893, 9361, 9841, 10333, 10837 The digital root of a star number is always 1 or 4, and progresses in the sequence 1, 4, 1. The last two digits of a star number in base 10 are always 01, 13, 21, 33, 37, 41, 53, 61, 73, 81, or 93. Unique among the star numbers is 35113, since its prime factors (i.e., 13, 37 and 73) are also consecutive star numbers. Relationships to other kinds of numbers Geometrically, the ''n''th star number is made up of a central point and 12 copies of the (''n''−1)th triangular n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Heptagonal Number
A centered heptagonal number is a centered figurate number that represents a heptagon with a dot in the center and all other dots surrounding the center dot in successive heptagonal layers. The centered heptagonal number for ''n'' is given by the formula :\over2. The first few centered heptagonal numbers are 1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, 547, 638, 736, 841, 953 Properties * Centered heptagonal numbers alternate parity in the pattern odd-even-even-odd. * A heptagonal numbers can expressed as a multiple of a triangular number by 7, plus one: :C_ = 7 * T_ + 1 *C_ is the sum of the integers between n+1 and 3n+1 (including) minus the sum of the integers from 0 to n (including). Centered heptagonal prime A centered heptagonal prime is a centered heptagonal number that is prime. The first few centered heptagonal primes are :43, 71, 197, 463, 547, 953, 1471, 1933, 2647, 2843, 3697, ... Due to parity, the centered heptagonal primes are in the fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Nonagonal Number
A centered nonagonal number (or centered enneagonal number) is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers. The centered nonagonal number for ''n'' layers is given by the formula :Nc(n) = \frac. Multiplying the (''n'' - 1)th triangular number by 9 and then adding 1 yields the ''n''th centered nonagonal number, but centered nonagonal numbers have an even simpler relation to triangular numbers: every third triangular number (the 1st, 4th, 7th, etc.) is also a centered nonagonal number. Thus, the first few centered nonagonal numbers are : 1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, 703, 820, 946. The list above includes the perfect numbers 28 and 496. All even perfect numbers are triangular numbers whose index is an odd Mersenne prime. Since every Mersenne prime greater than 3 is congruent to 1 modulo In computing, the modulo operation returns ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blum Integer
In mathematics, a natural number ''n'' is a Blum integer if is a semiprime for which ''p'' and ''q'' are distinct prime numbers congruent to 3 mod 4.Joe Hurd, Blum Integers (1997), retrieved 17 Jan, 2011 from http://www.gilith.com/research/talks/cambridge1997.pdf That is, ''p'' and ''q'' must be of the form , for some integer ''t''. Integers of this form are referred to as Blum primes. Goldwasser, S. and Bellare, M.br>"Lecture Notes on Cryptography". Summer course on cryptography, MIT, 1996-2001 This means that the factors of a Blum integer are Gaussian primes with no imaginary part. The first few Blum integers are : 21, 33, 57, 69, 77, 93, 129, 133, 141, 161, 177, 201, 209, 213, 217, 237, 249, 253, 301, 309, 321, 329, 341, 381, 393, 413, 417, 437, 453, 469, 473, 489, 497, ... The integers were named for computer scientist Manuel Blum. Properties Given a Blum integer, ''Q''''n'' the set of all quadratic residues modulo ''n'' and coprime to ''n'' and . Then: *''a'' h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aliquot Sequence
In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0. Definition and overview The aliquot sequence starting with a positive integer ''k'' can be defined formally in terms of the sum-of-divisors function σ1 or the aliquot sum function ''s'' in the following way: : ''s''0 = ''k'' : ''s''n = ''s''(''s''''n''−1) = σ1(''s''''n''−1) − ''s''''n''−1 if ''s''''n''−1 > 0 : ''s''n = 0 if ''s''''n''−1 = 0 ---> (if we add this condition, then the terms after 0 are all 0, and all aliquot sequences would be infinite sequence, and we can conjecture that all aliquot sequences are convergent, the limit of these sequences are usually 0 or 6) and ''s''(0) is undefined. For example, the aliquot sequence of 10 is 10, 8, 7, 1, 0 because: :σ1(10) − 10 = 5 + 2 + 1 = 8 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
