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Hexagonal Pyramidal Number
A pyramidal number is the number of points in a pyramid with a polygonal base and triangular sides. The term often refers to square pyramidal numbers, which have a square base with four sides, but it can also refer to a pyramid with any number of sides. The numbers of points in the base and in layers parallel to the base are given by polygonal numbers of the given number of sides, while the numbers of points in each triangular side is given by a triangular number. It is possible to extend the pyramidal numbers to higher dimensions. Formula The formula for the th -gonal pyramidal number is :P_n^r= \frac, where , . This formula can be factored: :P_n^r=\frac=\left(\frac\right)\left(\frac\right)=T_n \cdot \frac, where is the th triangular number. Sequences The first few triangular pyramidal numbers (equivalently, tetrahedral numbers) are: : 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ... The first few square pyramidal numbers are: : 1, 5, 14, 30, 55, 91, 140, 204, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Square Pyramidal Number
In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the stacked spheres in a pyramid (geometry), pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a broader topic of figurate numbers representing the numbers of points forming regular patterns within different shapes. As well as counting spheres in a pyramid, these numbers can be described algebraically as a sum of the first n positive square numbers, or as the values of a cubic polynomial. They can be used to solve several other counting problems, including counting squares in a square grid and counting acute triangles formed from the vertices of an odd regular polygon. They equal the sums of consecutive tetrahedral numbers, and are one-fourth of a larger tetrahedral number. The sum of two consecutive square pyramidal numbers is an octahedral number. History The pyramidal numbers were one of the few types of three-dimensional fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
14 (number)
14 (fourteen) is the natural number following 13 (number), 13 and preceding 15 (number), 15. Mathematics Fourteen is the seventh composite number. Properties 14 is the third distinct semiprime, being the third of the form 2 \times q (where q is a higher prime). More specifically, it is the first member of the second cluster of two discrete semiprimes (14, 15 (number), 15); the next such cluster is (21 (number), 21, 22 (number), 22), members whose sum is the fourteenth prime number, 43 (number), 43. 14 has an aliquot sum of 10 (number), 10, within an aliquot sequence of two composite numbers (14, 10 (number), 10, 8 (number), 8, 7 (number), 7, 1 (number), 1, 0) in the prime 7-aliquot tree. 14 is the third Pell number, companion Pell number and the fourth Catalan number. It is the lowest even n for which the Euler totient \varphi(x) = n has no solution, making it the first even nontotient. According to the Shapiro inequality, 14 is the least number n such that there exist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
196 (number)
196 (one hundred ndninety-six) is the natural number following 195 and preceding 197. In mathematics 196 is a square number, the square of 14. As the square of a Catalan number, it counts the number of walks of length 8 in the positive quadrant of the integer grid that start and end at the origin, moving diagonally at each step. It is part of a sequence of square numbers beginning 0, 1, 4, 25, 196, ... in which each number is the smallest square that differs from the previous number by a triangular number. There are 196 one-sided heptominoes, the polyominoes made from 7 squares. Here, one-sided means that asymmetric polyominoes are considered to be distinct from their mirror images. A Lychrel number is a natural number which cannot form a palindromic number through the iterative process of repeatedly reversing its digits and adding the resulting numbers. 196 is the smallest number conjectured to be a Lychrel number in base 10; the process has been carried out for over a billi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
126 (number)
126 (one hundred ndtwenty-six) is the natural number following 125 and preceding 127. In mathematics As the binomial coefficient \tbinom, 126 is a central binomial coefficient, and in Pascal's Triangle, it is a pentatope number. 126 is a sum of two cubes, and since 125 + 1 is σ3(5), 126 is the fifth value of the sum of cubed divisors function. 126 is the fifth \mathcal-perfect Granville number, and the third such not to be a perfect number. Also, it is known to be the smallest Granville number with three distinct prime factors, and perhaps the only such Granville number. 126 is a pentagonal pyramidal number and a decagonal number. 126 is also the different number of ways to partition a decagon into even polygons by diagonals, and the number of crossing points among the diagonals of a regular nonagon. There are exactly 126 binary strings of length seven that are not repetitions of a shorter string, and 126 different semigroups on four elements (up to is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
75 (number)
75 (seventy-five) is the natural number following 74 (number), 74 and preceding 76 (number), 76. __TOC__ In mathematics 75 is a self number because there is no integer that added up to its own digits adds up to 75. It is the sum of the first five pentagonal numbers, and therefore a pentagonal pyramidal number, as well as a nonagonal number. It is also the fourth ordered Bell number, and a Keith number, because it recurs in a Fibonacci-like sequence started from its base 10 digits: 7 (number), 7, 5 (number), 5, 12 (number), 12, 17 (number), 17, 29 (number), 29, 46 (number), 46, 75... 75 is the count of the number of weak orderings on a set of four items. Excluding the infinite sets, there are 75 uniform polyhedra in the third dimension, which incorporate Star polyhedron, star polyhedra as well. Inclusive of 7 families of Prism (geometry), prisms and antiprism, antiprisms, there are also 75 uniform polyhedron compound, uniform compound polyhedra. References Integers {{N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
40 (number)
40 (forty) is the natural number following 39 and preceding 41. Though the word is related to ''four'' (4), the spelling ''forty'' replaced ''fourty'' during the 17th century and is now the standard form. Mathematics 40 is an abundant number. Swiss mathematician Leonhard Euler noted 40 prime numbers generated by the quadratic polynomial n^ + n + 41, with values n = 0,1,2,...,39. These forty prime numbers are the same prime numbers that are generated using the polynomial n^ - n + 41 with values of n from 1 through 40, and are also known in this context as ''Euler's "lucky" numbers''. Forty is the only integer whose English name has its letters in alphabetical order. In religion The number 40 is found in many traditions without any universal explanation for its use. In Jewish, Christian, Islamic, and other Middle Eastern traditions it is taken to represent a large, approximate number, similar to " umpteen". Sumerian Enki ( /ˈɛŋki/) or Enkil (Sumerian: dEN.KI(G)𒂠... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
18 (number)
18 (eighteen) is the natural number following 17 (number), 17 and preceding 19 (number), 19. It is an even composite number. Mathematics 18 is a semiperfect number and an abundant number. It is a largely composite number, as it has 6 divisors and no smaller number has more than 6 divisors. There are 18 One-sided polyomino, one-sided pentominoes. In the classification of finite simple groups, there are 18 infinite families of groups. In science Chemistry * The 18-Electron rule, 18-electron rule is a rule of thumb in transition metal chemistry for characterising and predicting the stability of Metal complex#Metal complexes, metal complexes. In religion and literature * The Hebrew language, Hebrew word for "life" is (''Chai (symbol), chai''), which has a gematria, numerical value of 18. Consequently, the custom has arisen in Jewish circles to give donations and monetary gifts in multiples of 18 as an expression of blessing for long life. * In Judaism, in the Talmud; Pirkei Avot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
6 (number)
6 (six) is the natural number following 5 and preceding 7. It is a composite number and the smallest perfect number. In mathematics A six-sided polygon is a hexagon, one of the three regular polygons capable of tiling the plane. A hexagon also has 6 edges as well as 6 internal and external angles. 6 is the second smallest composite number. It is also the first number that is the sum of its proper divisors, making it the smallest perfect number. It is also the only perfect number that doesn't have a digital root of 1. 6 is the first unitary perfect number, since it is the sum of its positive proper unitary divisors, without including itself. Only five such numbers are known to exist. 6 is the largest of the four all-Harshad numbers. 6 is the 2nd superior highly composite number, the 2nd colossally abundant number, the 3rd triangular number, the 4th highly composite number, a pronic number, a congruent number, a harmonic divisor number, and a semiprime. 6 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
300 (number)
300 (three hundred) is the natural number following 299 and preceding 301. In Mathematics 300 is a composite number and the 24th triangular number. It is also a second hexagonal number. Integers from 301 to 399 300s 301 302 303 304 305 306 307 308 309 310s 310 311 312 313 314 315 315 = 32 × 5 × 7 = D_ \!, rencontres number, highly composite odd number, having 12 divisors. It is a Harshad number, as it is divisible by the sum of its digits. It is a Zuckerman number, as it is divisible by the product of its digits. 316 316 = 22 × 79, a centered triangular number and a centered heptagonal number. 317 317 is a prime number, Eisenstein prime with no imaginary part, Chen prime, one of the rare primes to be both right and left-truncatable, and a strictly non-palindromic number. 317 is the exponent (and number of ones) in the fourth base-10 repunit prime. 318 319 319 = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
280 (number)
280 (two hundred ndeighty) is the natural number after 279 and before 281. In mathematics The denominator of the eighth harmonic number, 280 is an octagonal number. 280 is the smallest octagonal number that is a half of another octagonal number. There are 280 plane tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, e.g., including only woody plants with secondary growth, only ...s with ten nodes. As a consequence of this, 18 people around a round table can shake hands with each other in non-crossing ways, in 280 different ways (this includes rotations). References {{DEFAULTSORT:280 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
204 (number)
204 (two hundred ndfour) is the natural number following 203 and preceding 205. In mathematics 204 is a refactorable number. 204 is a square pyramidal number: 204 balls may be stacked in a pyramid whose base is an 8 × 8 square. Its square, 2042 = 41616, is the fourth square triangular number. As a figurate number, 204 is also a nonagonal number and a truncated triangular pyramid number. 204 is a member of the Mian-Chowla sequence. There are exactly 204 irreducible quintic polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...s over a four-element field, exactly 204 ways to place three non-attacking chess queens on a 5 × 5 board, exactly 204 squares of an infinite chess move that are eight knight's moves from the center, exactly 204 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
140 (number)
140 (one hundred [and] forty) is the natural number following 139 (number), 139 and preceding 141 (number), 141. In mathematics 140 is an abundant number and a harmonic divisor number. It is the sum of the squares of the first seven integers, which makes it a square pyramidal number. 140 is an odious number because it has an odd number of ones in its binary representation. The sum of Euler's totient function φ(''x'') over the first twenty-one integers is 140. 140 is a repdigit in bases 13, 19, 27, 34, 69, and 139. In other fields 140 is also: * The former Twitter entry-character limit, a well-known characteristic of the service (based on the text messaging limit) ** A film, based on the Twitter entry-character limit, created and edited by Frank Kelly of Ireland References External links The Natural Number 140 {{DEFAULTSORT:140 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
