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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
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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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139 (number)
139 (one hundred [and] thirty-nine) is the natural number following 138 (number), 138 and preceding 140 (number), 140. In mathematics 139 is the 34th prime number. It is a twin prime with 137 (number), 137. Because 141 (number), 141 is a semiprime, 139 is a Chen prime. 139 is the smallest prime before a prime gap of length 10. This number is the sum of five consecutive prime numbers (19 (number), 19 + 23 (number), 23 + 29 (number), 29 + 31 (number), 31 + 37 (number), 37). It is the smallest factorization, factor of 64079 (number), 64079 which is the smallest Lucas number with prime index which is not prime. It is also the smallest factor of the first nine terms of the Euclid–Mullin sequence, making it the tenth term. 139 is a happy number and a strictly non-palindromic number. References External links Psalm 139
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141 (number)
141 (one hundred [and] forty-one) is the natural number following 140 (number), 140 and preceding 142 (number), 142. In mathematics 141 is: *a centered pentagonal number. *the sum of the sums of the divisors of the first 13 positive integers. *the second ''n'' to give a prime Cullen number (of the form ''n''2''n'' + 1). *an undulating number in base 10, with the previous being 131 (number), 131, and the next being 151 (number), 151. *the sixth hendecapolygonal number, gonal (11-gonal) number. *a semiprime: a product of two prime numbers, namely 3 and 47. Since those prime factors are Gaussian primes, this means that 141 is a Blum integer. * a Hilbert number, Hilbert prime References {{DEFAULTSORT:141 (Number) Integers ...
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Abundant Number
In number theory, an abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4, for example. Definition An ''abundant number'' is a natural number for which the Divisor function, sum of divisors satisfies , or, equivalently, the sum of proper divisors (or aliquot sum) satisfies . The ''abundance'' of a natural number is the integer (equivalently, ). Examples The first 28 abundant numbers are: :12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, ... . For example, the proper divisors of 24 are 1, 2, 3, 4, 6, 8, and 12, whose sum is 36. Because 36 is greater than 24, the number 24 is abundant. Its abundance is 36 − 24&nb ...
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Harmonic Divisor Number
In mathematics, a harmonic divisor number or Ore number is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are : 1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 . Harmonic divisor numbers were introduced by Øystein Ore, who showed that every perfect number is a harmonic divisor number and conjectured that there are no odd harmonic divisor numbers other than 1. Examples The number 6 has the four divisors 1, 2, 3, and 6. Their harmonic mean is an integer: \frac=2. Thus 6 is a harmonic divisor number. Similarly, the number 140 has divisors 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. Their harmonic mean is \frac=5. Since 5 is an integer, 140 is a harmonic divisor number. Factorization of the harmonic mean The harmonic mean of the divisors of any number can be expressed as the formula H(n) = \frac where is the sum of th powers of the divisors of : is the number of divisors, and is the s ...
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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 ...
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Euler's Totient Function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In other words, it is the number of integers in the range for which the greatest common divisor is equal to 1. The integers of this form are sometimes referred to as totatives of . For example, the totatives of are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since and . Therefore, . As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring \Z/n\Z). It is also used for defining the ...
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Repdigit
In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of "repeated" and "digit". Examples are 11, 666, 4444, and 999999. All repdigits are palindromic numbers and are multiples of repunits. Other well-known repdigits include the repunit primes and in particular the Mersenne primes (which are repdigits when represented in binary). Any such number can be represented as follows \underbrace_ = \frac Where nn is the concatenation of n with n. k the number of concatenated n. nn can be represented mathematically as n\cdot\left(10^+1\right) for n = 23 and k = 5, the formula will look like this \frac = \frac = \underbrace_ However, 2323232323 is not a repdigit. Also, any number can be decomposed into the sum and difference of the repdigit numbers. For example 3453455634 = 3333333333 + (111111111 + (99999 ...
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Twitter
Twitter, officially known as X since 2023, is an American microblogging and social networking service. It is one of the world's largest social media platforms and one of the most-visited websites. Users can share short text messages, images, and videos in Microblogging, short posts commonly known as "Tweet (social media), tweets" (officially "posts") and Like button, like other users' content. The platform also includes direct message, direct messaging, video and audio calling, bookmarks, lists, communities, a chatbot (Grok (chatbot), Grok), job search, and Spaces, a social audio feature. Users can vote on context added by approved users using the Community Notes feature. Twitter was created in March 2006 by Jack Dorsey, Noah Glass, Biz Stone, and Evan Williams (Internet entrepreneur), Evan Williams, and was launched in July of that year. Twitter grew quickly; by 2012 more than 100 million users produced 340 million daily tweets. Twitter, Inc., was based in San Francisco, C ...
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