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Pentagonal Pyramidal Number
A pyramidal number is a figurate number that represents a pyramid with a polygonal base and a given number of triangular sides. A pyramidal number is the number of points in a pyramid where each layer of the pyramid is an -sided polygon of points. The term often refers to square pyramidal numbers, which have a square base with four sides, but it can also refer to pyramids with three or more sides. The numbers of points in the base (and in parallel layers 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 numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Pyramidal Number
In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the number of stacked spheres in a 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 figurate num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5 (number)
5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number. It has attained significance throughout history in part because typical humans have five digits on each hand. In mathematics 5 is the third smallest prime number, and the second super-prime. It is the first safe prime, the first good prime, the first balanced prime, and the first of three known Wilson primes. Five is the second Fermat prime and the third Mersenne prime exponent, as well as the third Catalan number, and the third Sophie Germain prime. Notably, 5 is equal to the sum of the ''only'' consecutive primes, 2 + 3, and is the only number that is part of more than one pair of twin primes, ( 3, 5) and (5, 7). It is also a sexy prime with the fifth prime number and first prime repunit, 11. Five is the third factorial prime, an alternating factorial, and an Eisenstein prime with no imaginary part and real part of the for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 a pentatope number. It is also a decagonal number, a Harshad number and a pentagonal pyramidal number. As 125 + 1 it is σ3(5), the fifth value of the sum of cubed divisors function, and is a sum of two cubes. There are exactly 126 crossing points among the diagonals of a regular nonagon, 126 binary strings of length seven that are not repetitions of a shorter string, 126 different semigroups on four elements (up to isomorphism and reversal), and 126 different ways to partition a decagon into even polygons by diagonals. There are exactly 126 positive integers that are not solutions of the equation :x=abc+abd+acd+bcd, where ''a'', ''b'', ''c'', and ''d'' must themselves all be positive integers. It is the fifth Granville number, and the third such not to be a perfect number. Also, it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
75 (number)
75 (seventy-five) is the natural number following 74 and preceding 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, 5, 12, 17, 29, 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 polyhedra as well. Inclusive of 7 families of prisms and antiprisms, there are also 75 uniform compound polyhedra. In other fields Seventy-five is: *The atomic number of rhenium *The age limit for Canadian senators [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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" in the course of the 17th century and is now the standard form. In mathematics *Forty is a composite number, a refactorable number, an octagonal number, and—as the sum of the first four pentagonal numbers: 1 + 5 + 12 + 22 =40—it is a pentagonal pyramidal number. Adding up some subsets of its divisors (e.g., 1, 4, 5, 10, and 20) gives 40; hence, 40 is a semiperfect number. *Given 40, the Mertens function returns 0. 40 is the smallest number with exactly nine solutions to the equation Euler's totient function \varphi (x)=n. *Forty is the number of -queens problem solutions for n=7. *Forty is a repdigit in ternary (1111, ''i.e.'', 3^ + 3^ + 3^ + 3^, or, in other words, \frac ) and a Harshad number in decimal. In science *The atomic number of zirconium. *Negative forty is the unique temperature at which the Fahrenheit and Celsius ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
18 (number)
18 (eighteen) is the natural number following 17 and preceding 19. In mathematics * Eighteen is a composite number, its divisors being 1, 2, 3, 6 and 9. Three of these divisors (3, 6 and 9) add up to 18, hence 18 is a semiperfect number. Eighteen is the first inverted square-prime of the form ''p''·''q''2. * In base ten, it is a Harshad number. * It is an abundant number, as the sum of its proper divisors is greater than itself (1+2+3+6+9 = 21). It is known to be a solitary number, despite not being coprime to this sum. * It is the number of one-sided pentominoes. * It is the only number where the sum of its written digits in base 10 (1+8 = 9) is equal to half of itself (18/2 = 9). * It is a Fine number. In science Chemistry * Eighteen is the atomic number of argon. * Group 18 of the periodic table is called the noble gases. * The 18-electron rule is a rule of thumb in transition metal chemistry for characterising and predicting the stability of metal complexes. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Six is the smallest positive integer which is neither a square number nor a prime number; it is the second smallest composite number, behind 4; its proper divisors are , and . Since 6 equals the sum of its proper divisors, it is a perfect number; 6 is the smallest of the perfect numbers. It is also the smallest Granville number, or \mathcal-perfect number. As a perfect number: *6 is related to the Mersenne prime 3, since . (The next perfect number is 28 (number), 28.) *6 is the only even perfect number that is not the sum of successive odd cubes. *6 is the root of the 6-aliquot tree, and is itself the aliquot sum of only one other number; the square number, . Six is the only number that is both the sum and the product of three consecutive positive numbers. Unrelated to 6's being a perfect number, a Golomb ruler of length 6 is a "perfect ruler". Si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
300 (number)
300 (three hundred) is the natural number following 299 and preceding 301. Mathematical properties The number 300 is a triangular number and the sum of a pair of twin primes (149 + 151), as well as the sum of ten consecutive primes (13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47). It is palindromic in 3 consecutive bases: 30010 = 6067 = 4548 = 3639, and also in base 13. Factorization is 30064 + 1 is prime Other fields Three hundred is: * In bowling, a perfect score, achieved by rolling strikes in all ten frames (a total of twelve strikes) * The lowest possible Fair Isaac credit score * Three hundred ft/s is the maximum legal speed of a shot paintball * In the Hebrew Bible, the size of the military force deployed by the Israelite judge Gideon against the Midianites () * According to Islamic tradition, 300 is the number of ancient Israeli king Thalut's soldiers victorious against Goliath's soldiers * According to Herodotus, 300 is the number of ancient Spar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 trees 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). In geography *List of highways numbered 280 See also the year 280. Integers from 281 to 289 281 282 282 = 2·3·47, sphenic number, number of planar partitions of 9 283 283 prime, twin prime with 281, strictly non-palindromic number, 4283 - 3283 is prime 284 284 and 220 form the first pair of amicable numbers, as the divisors of 284 add up to 220 and vice versa. 285 285 = 3·5·19, sphenic number, square pyramidal number, Harshad number, repdigit in base 7 (555), vertically symmetric number , also in ''Star Trek'', the total nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 polynomials 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 strings of length 11 over a three-letter alphabet with no consecutively-repeated substring, and exactly 204 ways of immersing an oriented circle into the oriented plane so that it has four double points. Both 204 and its square are sums of a pair of twin p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
140 (number)
140 (one hundred ndforty) is the natural number following 139 and preceding 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 number of varieties of ashes from different varieties of pipe, cigar, and cigarette tobacco included in the Sherlock Holmes monograph. * 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 * The age that Job died at * The atomic number of unquadnilium, a temporary chemical element * PRO 140 anti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
91 (number)
91 (ninety-one) is the natural number following 90 and preceding 92. In mathematics 91 is: * the twenty-seventh distinct semiprime and the second of the form (7×q). * a triangular number. * a hexagonal number, one of the few such numbers to also be a centered hexagonal number. * a centered nonagonal number. * a centered cube number. * a square pyramidal number, being the sum of the squares of the first six integers. * the smallest positive integer expressible as a sum of two cubes in two different ways if negative roots are allowed (alternatively the sum of two cubes and the difference of two cubes): . (See 1729 for more details). This implies that 91 is the second cabtaxi number. * the smallest positive integer expressible as a sum of six distinct squares: . * The only other ways to write 91 as a sum of distinct squares are: and * . * the smallest pseudoprime satisfying the congruence .Friedman, ErichWhat's Special About This Number? * a repdigit in base 9 (1119). * palindr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
