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Centered Octagonal Number
A centered octagonal number is a centered number, centered figurate number that represents an octagon with a dot in the center and all other dots surrounding the center dot in successive octagonal layers.. The centered octagonal numbers are the same as the odd number, odd square numbers. Thus, the ''n''th odd square number and ''t''th centered octagonal number is given by the formula :O_n=(2n-1)^2 = 4n^2-4n+1 , (2t+1)^2=4t^2+4t+1. The first few centered octagonal numbers are :1 (number), 1, 9 (number), 9, 25 (number), 25, 49 (number), 49, 81 (number), 81, 121 (number), 121, 169 (number), 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089 (number), 1089, 1225 Calculating Ramanujan's tau function on a centered octagonal number yields an odd number, whereas for any other number the function yields an even number. O_n is the number of 2x2 matrices with elements from 0 to n that their determinant is twice their Permanent (mathematics), permanent. See also * Octagonal number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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81 (number)
81 (eighty-one) is the natural number following 80 (number), 80and preceding 82 (number), 82. In mathematics 81 is: * the square number, square of 9 (number), 9 and the second fourth-power of a prime; 34. * with an aliquot sum of 40 (number), 40; within an aliquot sequence of three composite numbers (81,40 (number), 40,50 (number), 50,43 (number), 43,1 (number), 1,0) to the Prime in the 43 (number), 43-aliquot tree. * a perfect totient number like all Power of three, powers of three. * a heptagonal number. * an Polygonal number, icosioctagonal number. * a centered octagonal number. * a tribonacci number. * an open meandric number. * the ninth member of the Mian-Chowla sequence. * a palindromic number in bases 8 (1218) and 26 (3326). * a Harshad number in bases 2, 3, 4, 7, 9, 10 and 13. * one of three non-trivial numbers (the other two are 1458 (number), 1458 and 1729 (number), 1729) which, when its digits (in decimal) are added together, produces a sum which, when multiplied by its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Permanent (mathematics)
In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix. Both are special cases of a more general function of a matrix called the immanant. Definition The permanent of an matrix is defined as \operatorname(A)=\sum_\prod_^n a_. The sum here extends over all elements σ of the symmetric group ''S''''n''; i.e. over all permutations of the numbers 1, 2, ..., ''n''. For example, \operatorname\begina&b \\ c&d\end=ad+bc, and \operatorname\begina&b&c \\ d&e&f \\ g&h&i \end=aei + bfg + cdh + ceg + bdi + afh. The definition of the permanent of ''A'' differs from that of the determinant of ''A'' in that the signatures of the permutations are not taken into account. The permanent of a matrix A is denoted per ''A'', perm ''A'', or Per ''A'', sometimes with parentheses around the argument. Minc uses Per(''A'') for the permanent of rectangula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis (linear algebra), basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible matrix, invertible and the corresponding linear map is an linear isomorphism, isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse. The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries. The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ramanujan's Tau Function
The Ramanujan tau function, studied by , is the function \tau : \mathbb\to\mathbb defined by the following identity: :\sum_\tau(n)q^n=q\prod_\left(1-q^n\right)^ = q\phi(q)^ = \eta(z)^=\Delta(z), where q=\exp(2\pi iz) with \mathrm(z)>0, \phi is the Euler function, \eta is the Dedekind eta function, and the function \Delta(z) is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form (some authors, notably Apostol, write \Delta/(2\pi)^ instead of \Delta). It appears in connection to an "error term" involved in counting the number of ways of expressing an integer as a sum of 24 squares. A formula due to Ian G. Macdonald was given in . Values The first few values of the tau function are given in the following table : Calculating this function on an odd square number (i.e. a centered octagonal number) yields an odd number, whereas for any other number the function yields an even number. Ramanujan's conjectures observed, but did not prove, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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1089 (number)
1089 is the integer after 1088 and before 1090. It is a square number (33 squared), a nonagonal number, a 32-gonal number, a 364-gonal number, and a centered octagonal number. 1089 is the first reverse-divisible number. The next is 2178 , and they are the only four-digit numbers that divide their reverse. In magic 1089 is widely used in magic tricks because it can be "produced" from any two three-digit numbers. This allows it to be used as the basis for a Magician's Choice. For instance, one variation of the book test starts by having the spectator choose any two suitable numbers and then apply some basic maths to produce a single four-digit number. That number is always 1089. The spectator is then asked to turn to page 108 of a book and read the 9th word, which the magician has memorized. To the audience it looks like the number is random, but through manipulation, the result is always the same. In base 10, the following steps always yield 1089: # Take any three-digit num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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169 (number)
169 (one hundred ndsixty-nine) is the natural number following 168 and preceding 170. In mathematics 169 is an odd number, a composite number, and a deficient number. 169 is a square number: 13 × 13 = 169, and if each number is reversed the equation is still true: 31 × 31 = 961. 144 shares this property: 12 × 12 = 144, 21 × 21 = 441. 169 is one of the few squares to also be a centered hexagonal number. Like all odd squares, it is a centered octagonal number. 169 is an odd-indexed Pell number, thus it is also a Markov number, appearing in the solutions (2, 169, 985), (2, 29, 169), (29, 169, 14701), etc. 169 is the sum of seven consecutive primes: 13 + 17 + 19 + 23 + 29 + 31 + 37. 169 is a difference in consecutive cubes, equaling 8^3-7^3. In other fields * 169 is known in the computing world as the first number of an automatic IPv4 address assigned by TCP/IP The Internet protocol suite, commonly known as TCP/IP, is a framework for organizing the commun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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121 (number)
121 (one hundred ndtwenty-one) is the natural number following 120 and preceding 122. In mathematics ''One hundred ndtwenty-one'' is * a square (11 times 11) * the sum of the powers of 3 from 0 to 4, so a repunit in ternary. Furthermore, 121 is the only square of the form 1 + p + p^2 + p^3 + p^4, where ''p'' is prime (3, in this case). * the sum of three consecutive prime numbers (37 + 41 + 43). * As 5! + 1 = 121, it provides a solution to Brocard's problem. There are only two other squares known to be of the form n! + 1. Another example of 121 being one of the few numbers supporting a conjecture is that Fermat conjectured that 4 and 121 are the only perfect squares of the form x^-4 (with being 2 and 5, respectively).Wells, D., '' The Penguin Dictionary of Curious and Interesting Numbers'', London: Penguin Group. (1987): 136 * It is also a star number, a centered tetrahedral number, and a centered octagonal number. * In decimal, it is a Smith number since its digits ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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25 (number)
25 (twenty-five) is the natural number following 24 and preceding 26. In mathematics It is a square number, being 52 = 5 × 5, and hence the third non-unitary square prime of the form ''p''2. It is one of two two-digit numbers whose square and higher powers of the number also ends in the same last two digits, e.g., 252 = 625; the other is 76. 25 has an even aliquot sum of 6, which is itself the first even and perfect number root of an aliquot sequence; not ending in ( 1 and 0). It is the smallest square that is also a sum of two (non-zero) squares: 25 = 32 + 42. Hence, it often appears in illustrations of the Pythagorean theorem. 25 is the sum of the five consecutive single-digit odd natural numbers 1, 3, 5, 7, and 9. 25 is a centered octagonal number, a centered square number, a centered octahedral number, and an automorphic number. 25 percent (%) is equal to . It is the smallest decimal Friedman number as it can be expressed by its own digits: 52. It ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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49 (number)
49 (forty-nine) is the natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ... following 48 and preceding 50. In mathematics Forty-nine is the square of the prime number seven and hence the fourth non-unitary square prime of the form ''p''2. Both of its digits are square numbers, 4 being the square of 2 and 9 being the square of 3. It appears in the Padovan sequence, preceded by the terms 21, 28, 37 (it is the sum of the first two of these). Along with the number that immediately derives from it, 77, the only number under 100 not having its home prime known (). The smallest triple of three squares in arithmetic succession is (1,25,49), and the second smallest is (49,169,289). 49 is the smallest discriminant of a totally real cubic field. 49 and 94 are the onl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Centered Number
In mathematics, the centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers of dots with a constant number of sides. Each side of a polygonal layer contains one more dot than each side in the previous layer; so starting from the second polygonal layer, each layer of a centered ''k''-gonal number contains ''k'' more dots than the previous layer. Examples Each centered ''k''-gonal number in the series is ''k'' times the previous triangular number, plus 1. This can be formalized by the expression \frac +1, where ''n'' is the series rank, starting with 0 for the initial 1. For example, each centered square number in the series is four times the previous triangular number, plus 1. This can be formalized by the expression \frac +1. These series consist of the * centered triangular numbers 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, ... (), * centered square numbers 1, 5, 13, 25, 41, 61, 85, 113, 145, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |