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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. 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 positive) is the previous regular triangular number. *Each centered triangular number from 10 onwards is the sum of three consecutive regular triangular ... [...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] |
31 (number)
31 (thirty-one) is the natural number following thirty, 30 and preceding 32 (number), 32. It is a prime number. In 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 a lucky prime and a happy number; two properties it shares with 13 (number), 13, which is its dual emirp and permutable prime. 31 is also a primorial prime, like its twin prime, 29 (number), 29. 31 is the number of regular polygons with an odd number of sides that are known to be constructible polygon, constructible with compass and straightedge, from combinations of known Fermat primes of the form 22''n'' + 1. 31 is the third Mersenne prime of the form 2''n'' − 1. It is also the eighth Mersenne prime exponent, specifically for the number 2,147,483,647, which is the maximum positive value for a 32-bit Integer (computer science), signed binary integer in computing. After 3, it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
274 (number)
270 (two hundred ndseventy) is the natural number following 269 and preceding 271. In mathematics *270 is a harmonic divisor number *270 is the fourth number that is divisible by its average integer divisor *270 is a practical number, by the second definition *The sum of the coprime counts for the first 29 integers is 270 *270 is a sparsely totient number, the largest integer with 72 as its totient *Given 6 elements, there are 270 square permutations *10! has 270 divisors *270 is a Harshad number in base 10 *270 is the smallest positive integer that has divisors ending by digits 1, 2, ..., 9. *270 is the smallest sum of a set of even numbers that contain every digit once. In other fields *The year 270 BC *The year 270 AD *The caliber of the .270 Winchester rifle *The number of U.S. Electoral College votes needed to be elected President of the United States *The average number of days in human pregnancy Integers from 271 to 279 271 272 272 = 24·17, sum of four consecutive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
235 (number)
235 (two hundred ndthirty-five) is the integer following 234 and preceding 236. In mathematics 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. In science *U-235 is the fissile isotope of uranium used in the first atomic bombs. See also * List of highways numbered 235 * 235 film, 35 mm film in daylight-loading spools * Superscope 235, a motion picture film format *2.35 to 1 widescreen aspect ratio in anamorphic format Anamorphic format is the cinem ... [...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. It is an emirp, given that the reversal of its digits is also prime (991). 199 is the smallest natural number that takes more than two iterations to compute its digital root 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 * The year AD 199 or 199 BC __NOTOC__ Year 199 BC was a year of the pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Lentulus and Tappulus (or, less frequently, year 555 '' Ab urbe condita''). The denomination 199 BC for this year has be ... * List of highways numbered 199 * References {{Integers, 1 Integers ... [...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. In astronomy * 166 Rhodope is a dark main belt asteroid, in the Adeona family of asteroids * 166P/NEAT is a periodic comet and centaur in the outer Solar System * HD 166 is the 6th magnitude star in the constellation Andromeda In the military * 166th Signal Photo Company was the official photo unit in the 89th Division of George Patton's Third Army in World War II * Convoy ON-166 was the 166th of the numbered ON series of merchant ship convoys outbound from the British Isles to North America departing February 11, 1943 * Marine Medium Helicopter Squadron 166 is a United States Marine Corps helicopter * was a United States Coast Guard cutter during World War II * was a United States Navy yacht. She was the first ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
136 (number)
136 (one hundred ndthirty six) is the natural number following 135 and preceding 137. In mathematics 136 is itself a factor of the Eddington number. With a total of 8 divisors, 8 among them, 136 is a refactorable number. It is a composite number. 136 is a centered triangular number and a centered nonagonal number. The sum of the ninth row of Lozanić's triangle is 136. 136 is a self-descriptive number in base 4, and a repdigit in base 16. In base 10, the sum of the cubes of its digits is 1^3 + 3^3 + 6^3 = 244. The sum of the cubes of the digits of 244 is 2^3 + 4^3 + 4^3 = 136. 136 is a triangular number, because it's the sum of the first 16 positive integers. In the military * Force 136 branch of the British organization, the Special Operations Executive (SOE), in the South-East Asian Theatre of World War II * USNS ''Mission Soledad'' (T-AO-136) was a United States Navy ''Mission Buenaventura''-class fleet oiler during World War II * USS ''Admirable'' (AM-136) was a United ... [...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 a super-prime. The previous prime is 107, making them both twin primes. 109 is a centered triangular number. 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 108-digit cycle are 853211, the first six Fibonacci numbers in descending order. 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, and 109 squares on an infinite chessboard that can be reached by a knight within three moves. See also *109 (other) 109 ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
85 (number)
85 (eighty-five) is the natural number following 84 and preceding 86. In mathematics 85 is: * the product of two prime numbers (5 and 17), and is therefore a semiprime; specifically, the 24th biprime not counting perfect squares. Together with 86 and 87, it forms the second cluster of three consecutive biprimes. * an octahedral number. * a centered triangular number. * a centered square number. * a decagonal number. * the smallest number that can be expressed as a 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 triangles. * a Smith number in decimal. In astronomy * Messier object M85 is a magnitude 10.5 lenticular galaxy in the constellation Coma Berenices * NGC 85 is a galaxy in the constellation Andromeda * 85 Io is a large main belt asteroid * 85 Pegasi is a multiple star system in constellation of Pegasus * 85 Ceti is a variable star in the constellation of Cetus * 85D/Boethin i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
64 (number)
64 (sixty-four) is the natural number following 63 and preceding 65. In mathematics Sixty-four is the square of 8, the cube of 4, and the sixth power of 2. It is the smallest number with exactly seven divisors. It is the lowest positive power of two that is adjacent to neither a Mersenne prime nor a Fermat prime. 64 is the sum of Euler's totient function for the first fourteen integers. It is also a dodecagonal number and a centered triangular number. 64 is also the first whole number (greater than 1) that is both a perfect square and a perfect cube. Since it is possible to find sequences of 64 consecutive integers such that each inner member shares a factor with either the first or the last member, 64 is an ErdÅ‘s–Woods number. In base 10, no integer added up to its own digits yields 64, hence it is a self number. 64 is a superperfect number—a number such that σ(σ(''n'')) = 2''n''. 64 is the index of Graham's number in the rapidly growing sequence 3 ↑↑↑â ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
46 (number)
46 (forty-six) is the natural number following 45 and preceding 47. In mathematics Forty-six is * a Wedderburn-Etherington number, * an enneagonal number * a centered triangular number. * the number of parallelogram polyominoes with 6 cells. It is the sum of the 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 possible to find sequences of 46+1 consecutive integers such that each inner member shares a factor with either the first or the last member, 46 is an Erdős–Woods number. In science * The atomic number of palladium. * The number of human chromosomes. * The approximate molar mass of ethanol (46.07 g mol) Astronomy * Messier object M46, a magnitude 6.5 open cluster in the constellation Puppis. * The New General Cataloguebr>objectNGC 46, a star in the constellation Pisces. In music * Japanese idol group franchise Sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
19 (number)
19 (nineteen) is the natural number following 18 and preceding 20. It is a prime number. Mathematics 19 is the eighth prime number, and forms a sexy prime with 13, a twin prime with 17, and a cousin prime with 23. It is the third full reptend prime, the fifth central trinomial coefficient, and the seventh Mersenne prime exponent. It is also the second Keith number, and more specifically the first Keith prime. * 19 is the maximum number of fourth powers needed to sum up to any natural number, and in the context of Waring's problem, 19 is the fourth value of g(k). * The sum of the squares of the first 19 primes is divisible by 19. *19 is the sixth Heegner number. 67 and 163, respectively the 19th and 38th prime numbers, are the two largest Heegner numbers, of nine total. * 19 is the third centered triangular number as well as the third centered hexagonal number. : The 19th triangular number is 190, equivalently the sum of the first 19 non-zero integers, that is al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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