177 (number)
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177 (number)
177 (one hundred ndseventy-seven) is the natural number following 176 and preceding 178. In mathematics One hundred and seventy-seven is the eighth Leyland number, where 177 = 2^7 + 7^2. The fifty-seventh semiprime is 177 (after the square of 13), and it is the fifty-first semiprime with distinct prime factors. The magic constant M of the smallest full 3 \times 3 magic square consisting of distinct primes is 177: Where the central cell \text 59 = \tfrac \text represents the seventeenth prime number, and seventh super-prime; equal to the sum of all prime numbers up to 17, including one: 1 + 2 + 3 + 5 + 7 + 11 + 13 + 17 = 59. 177 is also an arithmetic number, whose \sigma_0 holds an integer arithmetic mean of 60 — it is the ''one hundred and nineteenth'' indexed member in this sequence, where \text 59 + 60 = 119. The first non-trivial 60-''gonal'' number is 177. 177 is the tenth Leonardo number, part of a sequence of numbers closely related to the Fibon ...
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Natural Number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like Number (sports), jersey ...
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120 (number)
120 (one hundred ndtwenty) is the natural number following 119 and preceding 121. In the Germanic languages, the number 120 was also formerly known as "one hundred". This "hundred" of six score is now obsolete but is described as the long hundred or great hundred in historical contexts. In mathematics 120 is * the factorial of 5, i.e., 5!=5\cdot 4\cdot 3\cdot 2\cdot 1. * the fifteenth triangular number, as well as the sum of the first eight triangular numbers, making it also a tetrahedral number. 120 is the smallest number to appear six times in Pascal's triangle (as all triangular and tetragonal numbers appear in it). Because 15 is also triangular, 120 is a doubly triangular number. 120 is divisible by the first five triangular numbers and the first four tetrahedral numbers. It is the eighth hexagonal number. * The 10th highly composite, the 5th superior highly composite, superabundant, and the 5th colossally abundant number. It is also a sparsely totient number. ...
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Polygonal Number
In mathematics, a polygonal number is a Integer, number that counts dots arranged in the shape of a regular polygon. These are one type of 2-dimensional figurate numbers. Polygonal numbers were first studied during the 6th century BC by the Ancient Greeks, who investigated and discussed properties of Pronic number, oblong, Triangular Number, triangular, and Square number, square numbers. Definition and examples The number 10 for example, can be arranged as a triangle (see triangular number): : But 10 cannot be arranged as a square (geometry), square. The number 9, on the other hand, can be (see square number): : Some numbers, like 36, can be arranged both as a square and as a triangle (see square triangular number): : By convention, 1 is the first polygonal number for any number of sides. The rule for enlarging the polygon to the next size is to extend two adjacent arms by one point and to then add the required extra sides between those points. In the following diagrams, ...
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Triviality (mathematics)
In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or a particularly simple object possessing a given structure (e.g., group (mathematics), group, topological space). The noun triviality usually refers to a simple technical aspect of some proof or definition. The origin of the term in mathematical language comes from the medieval Trivium (education), trivium curriculum, which distinguishes from the more difficult quadrivium curriculum. The opposite of trivial is nontrivial, which is commonly used to indicate that an example or a solution is not simple, or that a statement or a theorem is not easy to prove. Triviality does not have a rigorous definition in mathematics. It is Subjectivity and objectivity (philosophy), subjective, and often determined in a given situation by the knowledge and experience of those considering the case. Trivial and nontrivial solutions In mathematics, the term "trivial" is ofte ...
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Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be '' finite'', as in these examples, or '' ...
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Arithmetic Mean
In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results from an experiment, an observational study, or a Survey (statistics), survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics because it helps to distinguish it from other types of means, such as geometric mean, geometric and harmonic mean, harmonic. Arithmetic means are also frequently used in economics, anthropology, history, and almost every other academic field to some extent. For example, per capita income is the arithmetic average of the income of a nation's Human population, population. While the arithmetic mean is often used to report central tendency, central tendencies, it is not a robust statistic: it is greatly influenced by outliers (Value (mathematics), values much larger or smaller than ...
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Arithmetic Number
In number theory, an arithmetic number is an integer for which the average of its positive divisors is also an integer. For instance, 6 is an arithmetic number because the average of its divisors is :\frac=3, which is also an integer. However, 2 is not an arithmetic number because its only divisors are 1 and 2, and their average 3/2 is not an integer. The first numbers in the sequence of arithmetic numbers are :1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, ... . The arithmetic means of the divisors of arithmetic numbers are listed at . Density It is known that the natural density of such numbers is 1:Guy (2004) p.76 indeed, the proportion of numbers less than ''X'' which are not arithmetic is asymptotically :\exp\left( \,\right) where ''c'' = 2 + o(1). A number ''N'' is arithmetic if the number of divisors ''d''(''N'') divides the sum of divisors σ(''N''). It is known that the density De ...
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17 (number)
17 (seventeen) is the natural number following 16 (number), 16 and preceding 18 (number), 18. It is a prime number. 17 was described at MIT as "the least random number", according to the Jargon File. This is supposedly because, in a study where respondents were asked to choose a random number from 1 to 20, 17 was the most common choice. This study has been repeated a number of times. Mathematics 17 is a Leyland number and Leyland number#Leyland primes, Leyland prime, using 2 & 3 (23 + 32) and using 4 and 5, using 3 & 4 (34 - 43). 17 is a Fermat prime. 17 is one of six lucky numbers of Euler. Since seventeen is a Fermat prime, regular heptadecagons can be constructible polygon, constructed with a compass and unmarked ruler. This was proven by Carl Friedrich Gauss and ultimately led him to choose mathematics over philology for his studies. The minimum possible number of givens for a sudoku puzzle with a unique solution is 17. Geometric properties Two-dimensions *There are ...
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Super-prime
Super-prime numbers, also known as higher-order primes or prime-indexed primes (PIPs), are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers. In other words, if prime numbers are matched with ordinal numbers, starting with prime number 2 matched with ordinal number 1, then the primes matched with prime ordinal numbers are the super-primes. The subsequence begins :3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211, 241, 277, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991, ... . That is, if ''p''(''n'') denotes the ''n''th prime number, the numbers in this sequence are those of the form ''p''(''p''(''n'')). In 1975, Robert Dressler and Thomas Parker used a computer-aided proof (based on calculations involving the subset sum problem) to show that every integer greater than 96 may be represented as a sum of distinct super-prime numbers. Their proof ...
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Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ...
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496 (number)
496 (four hundred ndninety-six) is the natural number following 495 and preceding 497. In mathematics 496 is most notable for being a perfect number, and one of the earliest numbers to be recognized as such. 496 is also a harmonic divisor number. The group ''E''8 has real dimension 496. In physics The number 496 is a very important number in superstring theory. In 1984, Michael Green and John H. Schwarz realized that one of the necessary conditions for a superstring theory to make sense is that the dimension of the gauge group of type I string theory must be 496. The group is therefore SO(32). Their discovery started the first superstring revolution. It was realized in 1985 that the heterotic string can admit another possible gauge group, namely E8 x E8. Telephone numbers The UK's Ofcom The Office of Communications, commonly known as Ofcom, is the government-approved regulatory and competition authority for the broadcasting, internet, telecommunications an ...
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28 (number)
28 (twenty-eight) is the natural number following 27 (number), 27 and preceding 29 (number), 29. In mathematics Twenty-eight is a composite number and the second perfect number as it is the sum of its proper divisors: 1+2+4+7+14=28. As a perfect number, it is related to the Mersenne prime 7, since 2^\times (2^-1)=28. The next perfect number is 496 (number), 496, the previous being 6 (number), 6. Though perfect, 28 is not the aliquot sum of any other number other than itself; thus, it is not part of a multi-number aliquot sequence. Twenty-eight is the sum of the totient function for the first nine integers. Since the greatest prime factor of 28^+1=785 is 157, which is more than 28 twice, 28 is a Størmer number. Twenty-eight is a harmonic divisor number, a happy number, the 7th triangular number, a hexagonal number, a Leyland number#Leyland number of the second kind, Leyland number of the second kind (2^6-6^2), and a centered nonagonal number. It appears in the Padovan sequ ...
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