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Centered Tetrahedral Number
A centered tetrahedral number is a centered figurate number that represents a tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the .... The centered tetrahedral number for a specific ''n'' is given by (2n+1)\times The first such numbers are 1, 5, 15, 35, 69, 121, 195, 295, 425, 589, 791, ... . Parity and divisibility *Every centered tetrahedral number is odd. *Every centered tetrahedral number with an index of 2, 3 or 4 modulo 5 is divisible by 5. References * Figurate numbers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polygonal number * a number represented as a discrete -dimensional regular geometric pattern of -dimensional balls such as a polygonal number (for ) or a polyhedral number (for ). * a member of the subset of the sets above containing only triangular numbers, pyramidal numbers, and their analogs in other dimensions. Terminology Some kinds of figurate number were discussed in the 16th and 17th centuries under the name "figural number". In historical works about Greek mathematics the preferred term used to be ''figured number''. In a use going back to Jacob Bernoulli's Ars Conjectandi, the term ''figurate number'' is used for triangular numbers made up of successive integers, tetrahedral numbers made up of successive triangular numbers, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
15 (number)
15 (fifteen) is the natural number following 14 and preceding 16. Mathematics 15 is: * A composite number, and the sixth semiprime; its proper divisors being , and . * A deficient number, a smooth number, a lucky number, a pernicious number, a bell number (i.e., the number of partitions for a set of size 4), a pentatope number, and a repdigit in binary (1111) and quaternary (33). In hexadecimal, and higher bases, it is represented as F. * A triangular number, a hexagonal number, and a centered tetrahedral number. * The number of partitions of 7. * The smallest number that can be factorized using Shor's quantum algorithm. * The magic constant of the unique order-3 normal magic square. * The number of supersingular primes. Furthermore, * 15 is one of two numbers within the ''teen'' numerical range (13-19) not to use a single-digit number in the prefix of its name (the first syllable preceding the ''teen'' suffix); instead, it uses the adjective form of fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
35 (number)
35 (thirty-five) is the natural number following 34 and preceding 36. In mathematics 35 is the sum of the first five triangular numbers, making it a tetrahedral number. 35 is the number of ways that three things can be selected from a set of seven unique things, also known as the " combination of seven things taken three at a time". 35 is a centered cube number, a centered tetrahedral number, a pentagonal number, and a pentatope number. 35 is a highly cototient number, since there are more solutions to the equation x - \varphi (x) = 35 than there are for any other integers below it except 1. There are 35 free hexominoes, the polyominoes made from six squares. Since the greatest prime factor of 35^ + 1 = 1226 is 613, which is more than 35 twice, 35 is a Størmer number. 35 is the tenth discrete semiprime (5 \times 7) and the first with 5 as the lowest non-unitary factor. The aliquot sum of 35 is 13, this being the second composite number with such an aliquot sum; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
69 (number)
69 (sixty-nine) is the natural number following 68 and preceding 70. In mathematics 69 is: * a lucky number. * a semiprime. * a Blum integer, since the two factors of 69 are both Gaussian primes. * the sum of the sums of the divisors of the first 9 positive integers. * the third composite number in the 13-aliquot tree. The aliquot sum of sixty-nine is 27 within the aliquot sequence (69,27,13,1,0). * a strobogrammatic number. * a centered tetrahedral number. Because 69 has an odd number of 1s in its binary representation, it is sometimes called an " odious number." 69 is the only number whose square () and cube () use every decimal digit from 0–9 exactly once.David Wells: ''The Penguin Dictionary of Curious and Interesting Numbers'' 69 is equal to 105 octal, while 105 is equal to 69 hexadecimal. This same property can be applied to all numbers from 64 to 69. On many handheld scientific and graphing calculators, the highest factorial that can be calculated, due to m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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). * 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 add up to the same value as its factorization (which uses the same digits) and as a consequence of that it is a Friedm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
195 (number)
195 (one hundred ndninety-five) is the natural number following 194 and preceding 196. In mathematics 195 is: * the sum of eleven consecutive primes: 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 * the smallest number expressed as a sum of distinct squares in 16 different ways * a centered tetrahedral number * in the middle of a prime quadruplet In number theory, a prime quadruplet (sometimes called prime quadruple) is a set of four prime numbers of the form This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4. Prim ... (191, 193, 197, 199). See also * 195 (other) References {{DEFAULTSORT:195 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Figurate Number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polygonal number * a number represented as a discrete -dimensional regular geometric pattern of -dimensional balls such as a polygonal number (for ) or a polyhedral number (for ). * a member of the subset of the sets above containing only triangular numbers, pyramidal numbers, and their analogs in other dimensions. Terminology Some kinds of figurate number were discussed in the 16th and 17th centuries under the name "figural number". In historical works about Greek mathematics the preferred term used to be ''figured number''. In a use going back to Jacob Bernoulli's Ars Conjectandi, the term ''figurate number'' is used for triangular numbers made up of successive integers, tetrahedral numbers made up of successive triangular numbers, etc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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