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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 form ... [...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 form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Number
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, 181, 221, 265, ... ... [...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 geometry, geometric pattern of -dimensional Ball (mathematics), 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 successi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heptagon
In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of ''septua-'', a Latin-derived numerical prefix, rather than ''hepta-'', a Greek-derived numerical prefix; both are cognate) together with the Greek suffix "-agon" meaning angle. Regular heptagon A regular heptagon, in which all sides and all angles are equal, has internal angles of 5π/7 radians (128 degrees). Its Schläfli symbol is . Area The area (''A'') of a regular heptagon of side length ''a'' is given by: :A = \fraca^2 \cot \frac \simeq 3.634 a^2. This can be seen by subdividing the unit-sided heptagon into seven triangular "pie slices" with vertices at the center and at the heptagon's vertices, and then halving each triangle using the apothem as the common side. The apothem is half the cotangent of \pi/7, and the area of each of the 14 small triangles is one-fourth of the apothem. The area of a regular heptago ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1 (number)
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0. The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 equals the same number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention not considered a prime number; this was not universally accepted until the mid-20th century. Additionally, 1 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
8 (number)
8 (eight) is the natural number following 7 and preceding 9. In mathematics 8 is: * a composite number, its proper divisors being , , and . It is twice 4 or four times 2. * a power of two, being 2 (two cubed), and is the first number of the form , being an integer greater than 1. * the first number which is neither prime nor semiprime. * the base of the octal number system, which is mostly used with computers. In octal, one digit represents three bits. In modern computers, a byte is a grouping of eight bits, also called an wikt:octet, octet. * a Fibonacci number, being plus . The next Fibonacci number is . 8 is the only positive Fibonacci number, aside from 1, that is a perfect cube. * the only nonzero perfect power that is one less than another perfect power, by Catalan conjecture, Mihăilescu's Theorem. * the order of the smallest non-abelian group all of whose subgroups are normal. * the dimension of the octonions and is the highest possible dimension of a normed divisio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
22 (number)
22 (twenty-two) is the natural number following 21 and preceding 23. In mathematics 22 is a palindromic number and the eighth semiprime; its proper divisors are 1, 2, and 11. It is the second Smith number, the second Erdős–Woods number, and the fourth large Schröder number. It is also a Perrin number, from a sum of 10 and 12. 22 is the fourth pentagonal number, the third hexagonal pyramidal number, and the third centered heptagonal number. The maximum number of regions into which five intersecting circles divide the plane is 22. 22 is also the quantity of pieces in a disc that can be created with six straight cuts, which makes 22 the seventh central polygonal number. \frac is a commonly used approximation of the irrational number , the ratio of the circumference of a circle to its diameter; where both 22 and 7 are consecutive hexagonal pyramidal numbers. 22 also features in another approximation for pi, here by Srinivasa Ramanujan from an approximate constr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
43 (number)
43 (forty-three) is the natural number following 42 and preceding 44. In mathematics Forty-three is the 14th smallest prime number. The previous is forty-one, with which it comprises a twin prime, and the next is forty-seven. 43 is the smallest prime that is not a Chen prime. It is also the third Wagstaff prime. 43 is the fourth term of Sylvester's sequence, one more than the product of the previous terms (2 × 3 × 7). 43 is a centered heptagonal number. Let ''a'' = ''a'' = 1, and thenceforth ''a'' = (''a'' + ''a'' + ... + ''a''). This sequence continues 1, 1, 2, 3, 5, 10, 28, 154... . ''a'' is the first term of this sequence that is not an integer. 43 is a Heegner number. 43 is the largest prime which divides the order of the Janko group J4. 43 is a repdigit in base 6 (111). 43 is the number of triangles inside the Sri Yantra. 43 is the largest natural number that is not an (original) McNugget number. 43 is the smallest prime number expressible as the sum of 2, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
71 (number)
71 (seventy-one) is the natural number following 70 (number), 70 and preceding 72 (number), 72. __TOC__ In mathematics 71 is: *the 20th prime number. The next is 73 (number), 73, with which it composes a twin prime. *a permutable prime and emirp with 17 (number), 17. *is the largest number which occurs as a prime factor of an order of a sporadic simple group. *the sum of three consecutive primes: 19 (number), 19, 23 (number), 23 and 29 (number), 29. *a centered heptagonal number. *an Eisenstein prime with no imaginary part and real part of the form 3''n'' – 1. *a Pillai prime, since 9! + 1 is divisible by 71 but 71 is not one more than a multiple of 9. *the largest (15th) Supersingular prime (moonshine theory), supersingular prime, which is also a Chen prime. *part of the last known pair (71, 7) of Brown numbers, since 712 = 7! + 1. *the twenty-third term of the Euclid–Mullin sequence, as it is the least prime factor of one more than th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
106 (number)
106 (one hundred ndsix) is the natural number following 105 and preceding 107. In mathematics 106 is a centered pentagonal number, a 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 ..., and a regular 19-gonal number. There are 106 mathematical trees with ten vertices. See also * 106 (other) References {{DEFAULTSORT:106 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
148 (number)
148 (one hundred ndforty-eight) is the natural number following 147 and before 149. In mathematics 148 is the second number to be both a heptagonal number and a centered heptagonal number (the first is 1). It is the twelfth member of the Mian–Chowla sequence, the lexicographically smallest sequence of distinct positive integers with distinct pairwise sums. There are 148 perfect graphs with six vertices, and 148 ways of partitioning four people into subsets, ordering the subsets, and selecting a leader for each subset. In other fields In the Book of Nehemiah 7:44 there are 148 singers, sons of Asaph, at the census of men of Israel upon return from exile. This differs from Ezra 2:41, where the number is given as 128. Dunbar's number is a theoretical cognitive limit to the number of people with whom one can maintain stable interpersonal relationships. Dunbar predicted a "mean group size" of 148, but this is commonly rounded to 150. See also * The year AD 148 or 148 BC _ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
197 (number)
197 (one hundred ndninety-seven) is the natural number following 196 and preceding 198. In mathematics * 197 is a prime number, the third of a prime quadruplet: 191, 193, 197, 199 * 197 is the smallest prime number that is the sum of 7 consecutive primes: 17 + 19 + 23 + 29 + 31 + 37 + 41, and is the sum of the first twelve prime numbers: 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 * 197 is a centered heptagonal number, 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 * 197 is a Schröder–Hipparchus number, counting for instance the number of ways of subdividing a heptagon by a non-crossing set of its diagonals. In other fields 197 is also: * A police emergency telephone number in Tunisia * Number enquiry telephone number in Nepal * a song by Norwegian alternative rock group Major Parkinson Major Parkinson is a Norwegian rock group currently based in Berge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
