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Centered Triangular Number
A centered (or centred) triangular number is a centered figurate number that represents an equilateral triangle with a dot in the center and all its other dots surrounding the center in successive equilateral triangular layers. This is also the number of points of a hexagonal lattice with nearest-neighbor coupling whose distance from a given point is less than or equal to n. The following image shows the building of the centered triangular numbers by using the associated figures: at each step, the previous triangle (shown in red) is surrounded by a triangular layer of new dots (in blue). Properties *The gnomon of the ''n''-th centered triangular number, corresponding to the (''n'' + 1)-th triangular layer, is: ::C_ - C_ = 3(n+1). *The ''n''-th centered triangular number, corresponding to ''n'' layers ''plus'' the center, is given by the formula: ::C_ = 1 + 3 \frac = \frac. *Each centered triangular number has a remainder of 1 when divided by 3, and the quotient (if posi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
31 (number)
31 (thirty-one) is the natural number following thirty, 30 and preceding 32 (number), 32. It is a prime number. Mathematics 31 is the 11th prime number. It is a superprime and a Self number#Self primes, self prime (after 3, 5, and 7), as no integer added up to its base 10 digits results in 31. It is the third Mersenne prime of the form 2''n'' − 1, and the eighth Mersenne prime ''exponent'', in-turn yielding the maximum positive value for a 32-bit Integer (computer science), signed binary integer in computing: 2,147,483,647. After 3, it is the second Mersenne prime not to be a double Mersenne prime, while the 31st prime number (127 (number), 127) is the second double Mersenne prime, following 7. On the other hand, the thirty-first triangular number is the perfect number 496 (number), 496, of the form 2(5 − 1)(25 − 1) by the Euclid-Euler theorem. 31 is also a ''primorial prime'' like its twin prime (29 (number), 29), as well as both a lucky prime and a happy number like its d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
274 (number)
274 (two hundred ndseventy-four) is the natural number following 273 and preceding 275. In mathematics *274 is an even composite number. *274's sum of its proper divisors is 140. *The number 274 is the 13th tribonacci number. This is defined by the equations P(0)=P(1)=0 P(2)=1 and P(n)=P(n-1)+P(n-2)+P(n-3). *274 is the sum of 5 perfect cubes. It is the sum of 23+23+23+53+53. *274 is a Stirling number of the first kind which counts the number of permutations and their number of cycles Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in .... References {{Integers, 2 Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
235 (number)
235 (two hundred ndthirty-five) is the integer following 234 and preceding 236. Additionally, 235 is: *a semiprime *a heptagonal number *a centered triangular number *therefore a figurate number in two ways *palindromic in bases 4 (32234), 7 (4547), 8 (3538), 13 (15113), and 46 (5546) *a Harshad number in bases 6, 47, 48, 95, 116, 189 and 231 *a Smarandache–Wellin number Also: *There are 235 different trees with 11 unlabeled nodes. *If an equilateral triangle is subdivided into smaller equilateral triangles whose side length is 1/9 as small, the resulting "matchstick arrangement" will have exactly 235 different equilateral triangles of varying sizes in it. *The Metonic cycle is 235 synodic month In lunar calendars, a lunar month is the time between two successive Syzygy (astronomy), syzygies of the same type: new moons or full moons. The precise definition varies, especially for the beginning of the month. Variations In Shona people, S ...s. References Integers [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
199 (number)
199 (one hundred ndninety-nine) is the natural number following 198 and preceding 200. In mathematics 199 is a centered triangular number. It is a prime number and the fourth part of a prime quadruplet: 191, 193, 197, 199. 199 is the smallest natural number that takes more than two iterations to compute its digital root The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit su ... as a repeated digit sum: \begin 199&\mapsto 1+9+9=19\\ &\mapsto 1+9=10\\ &\mapsto 1+0=1. \end Thus, its additive persistence is three, and it is the smallest number of persistence three. See also * References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
166 (number)
166 (one hundred ndsixty-six) is the natural number following 165 and preceding 167. In mathematics 166 is an even number and a composite number. It is a centered triangular number. Given 166, the Mertens function returns 0. 166 is a Smith number in base 10. 166 in Roman numerals Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, eac ... consists of the first 5 symbols, CLXVI. External links Number Facts and Trivia: 166The Number 166 166th Street (3rd Avenue El) References {{DEFAULTSORT:166 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
136 (number)
136 (one hundred [and] thirty-six) is the natural number following 135 (number), 135 and preceding 137 (number), 137. In mathematics 136 is: * a refactorable number and a composite number. * the 16th triangular number. * a repdigit in base 16 (88). External links 136 cats(video) References {{DEFAULTSORT:136 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
109 (number)
109 (one hundred ndnine) is the natural number following 108 and preceding 110. In mathematics 109 is the 29th prime number. As 29 is itself prime, 109 is the tenth super-prime. The previous prime is 107, making them both twin primes. 109 is a centered triangular number. There are exactly: *109 different families of subsets of a three-element set whose union includes all three elements. *109 different loops (invertible but not necessarily associative binary operations with an identity) on six elements. *109 squares on an infinite chessboard that can be reached by a knight within three moves. There are 109 uniform edge-colorings to the 11 regular and semiregular (or Archimedean) tilings. The decimal expansion of 1/109 can be computed using the alternating series, with F(n) the n^ Fibonacci number: ::\frac=\sum_^\infty\times (-1)^=0.00917431\dots The decimal expansion of 1/109 has 108 digits, making 109 a full reptend prime in decimal. The last six digits of the 10 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
85 (number)
85 (eighty-five) is the natural number following 84 (number), 84 and preceding 86 (number), 86. In mathematics 85 is: * the product of two prime numbers (5 and 17), and is therefore a semiprime of the form (5.q) where q is prime. * specifically, the 24th Semiprime, it being the fourth of the form (5.q). *together with 86 (number) , 86 and 87 (number), 87, forms the second cluster of three consecutive semiprimes; the first comprising 33 (number), 33, 34 (number), 34, 35 (number), 35. * with a prime aliquot sum of 23 (number), 23 in the short aliquot sequence (85,23 (number), 23,1 (number), 1,0). * an octahedral number. * a centered triangular number. * a centered square number. * a decagonal number. * the smallest number that can be expressed as a Fermat's theorem on sums of two squares, sum of two squares, with all squares greater than 1, in two ways, 85 = 92 + 22 = 72 + 62. * the length of the hypotenuse of four pythagorean triple, Pythagorean triangles. * a Smith number in deci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
64 (number)
64 (sixty-four) is the natural number following 63 (number), 63 and preceding 65 (number), 65. Mathematics Sixty-four is the square of 8 (number), 8, the cube of 4 (number), 4, and the sixth power of 2 (number), 2. It is the seventeenth interprime, since it lies midway between the eighteenth and nineteenth prime numbers (61 (number), 61, 67 (number), 67). The aliquot sum of a power of two (2''n'') is always one less than the power of two itself, therefore the aliquot sum of 64 is 63 (number), 63, within an aliquot sequence of two composite members (64, 63 (number), 63, 41 (number), 41, 1 (number), 1, 0) that are rooted in the aliquot tree of the thirteenth prime, 41. 64 is: *the smallest number with exactly seven divisors, *the first whole number (greater than one) that is both a perfect square, and a perfect cube, *the lowest positive power of two that is not adjacent to either a Mersenne prime or a Fermat prime, *the fourth superperfect number — a number such that divisor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
46 (number)
46 (forty-six) is the natural number following 45 (number), 45 and preceding 47 (number), 47. In mathematics Forty-six is * thirteenth discrete semiprime (2 \times 23) and the eighth of the form (2.q), where q is a higher prime, * with an aliquot sum of 26 (number), 26; a semiprime, in an aliquot sequence of six composite numbers (46, 26,16 (number), 16, 15 (number), 15, 9 (number), 9, 4 (number), 4, 3 (number), 3, 1 (number), 1, 0) in the prime 3 (number), 3-aliquot tree, * a Wedderburn-Etherington number, * the second non-trivial nonagonal number, enneagonal number, after 24 (number), 24, * a centered triangular number, * the number of parallelogram polyominoes with 6 cells. * the amount of prime numbers in between 1 and 200 (number), 200. It is the sum of the Euler's totient function, totient function for the first twelve integers. 46 is the largest even integer that cannot be expressed as a sum of two abundant numbers. It is also the sixteenth semiprime. Since it is pos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
19 (number)
19 (nineteen) is the natural number following 18 (number), 18 and preceding 20 (number), 20. It is a prime number. Mathematics Nineteen is the eighth prime number. Number theory 19 forms a twin prime with 17 (number), 17, a cousin prime with 23 (number), 23, and a sexy prime with 13 (number), 13. 19 is the fifth Trinomial triangle#Central trinomial coefficients, central trinomial coefficient, and the maximum number of fourth powers needed to sum up to any natural number (see, Waring's problem). It is the number of Composition (combinatorics), compositions of 8 into distinct parts. 19 is the eighth strictly non-palindromic number in any Numeral system, base, following 11 (number), 11 and preceding 47 (number), 47. 19 is also the second octahedral number, after 6, and the sixth Heegner number. In the Engel expansion of pi, 19 is the seventh term following and preceding . The sum of the first terms preceding 17 (number), 17 is in equivalence with 19, where its prime Sequen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
