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Semiperfect
In number theory, a semiperfect number or pseudoperfect number is a natural number ''n'' that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. The first few semiperfect numbers are: 6, 12, 18, 20, 24, 28, 30, 36, 40, ... Properties * Every multiple of a semiperfect number is semiperfect. A semiperfect number that is not divisible by any smaller semiperfect number is called ''primitive''. * Every number of the form 2''m''''p'' for a natural number ''m'' and an odd prime number ''p'' such that ''p'' < 2''m''+1 is also semiperfect. ** In particular, every number of the form 2''m''(2''m''+1 − 1) is semiperfect, and indeed perfect if 2''m''+1 − 1 is a . * The sma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, since 1 + 2 + 4 + 7 + 14 = 28. The first four perfect numbers are 6 (number), 6, 28 (number), 28, 496 (number), 496 and 8128 (number), 8128. The sum of proper divisors of a number is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors; in symbols, \sigma_1(n)=2n where \sigma_1 is the sum-of-divisors function. This definition is ancient, appearing as early as Euclid's Elements, Euclid's ''Elements'' (VII.22) where it is called (''perfect'', ''ideal'', or ''complete number''). Euclid also proved a formation rule (IX.36) whereby \frac is an even perfect number whenever q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
304 (number)
304 is the natural number following 303 and preceding 305. In mathematics *304 is an even composite number with two prime factors. *304 is the sum of consecutive primes in two different ways: It is the sum of 41+43+47+53+59+61 and of 23+29+31+37+41+43+47+53. *304 is a primitive semiperfect number meaning that it is a semiperfect number that is not divisible by any other semiperfect number. *304 is an untouchable number meaning that it is not equal to the sum of any number’s proper divisors. *304 is a nontotient In number theory, a nontotient is a positive integer ''n'' which is not a totient number: it is not in the range of Euler's totient function φ, that is, the equation φ(''x'') = ''n'' has no solution ''x''. In other words, ''n'' is a nontotie ... number meaning that it is an even number where phi(x) cannot result in that number. References {{Improve categories, date=October 2023 Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weird Number
In number theory, a weird number is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself. Examples The smallest weird number is 70. Its proper divisors are 1, 2, 5, 7, 10, 14, and 35; these sum to 74, but no subset of these sums to 70. The number 12, for example, is abundant but ''not'' weird, because the proper divisors of 12 are 1, 2, 3, 4, and 6, which sum to 16; but 2 + 4 + 6 = 12. The first several weird numbers are : 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, ... . Properties Infinitely many weird numbers exist. For example, 70''p'' is weird for all primes ''p'' ≥ 149. In fact, the set of weird numbers has positive asymptotic density. It is not known if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
30 (number)
30 (thirty) is the natural number following 29 and preceding 31. In mathematics 30 is an even, composite, and pronic number. With 2, 3, and 5 as its prime factors, it is a regular number and the first sphenic number, the smallest of the form , where is a prime greater than 3. It has an aliquot sum of 42; within an aliquot sequence of thirteen composite numbers (30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0) to the Prime in the 3-aliquot tree. From 1 to the number 30, this is the longest Aliquot Sequence. It is also: * A semiperfect number, since adding some subsets of its divisors (e.g., 5, 10 and 15) equals 30. * A primorial. * A Harshad number in decimal. * Divisible by the number of prime numbers ( 10) below it. * The largest number such that all coprimes smaller than itself, except for 1, are prime. * The sum of the first four squares, making it a square pyramidal number. * The number of vertices in the Tutte–Coxeter graph. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
12 (number)
12 (twelve) is the natural number following 11 (number), 11 and preceding 13 (number), 13. Twelve is the 3rd superior highly composite number, the 3rd colossally abundant number, the 5th highly composite number, and is divisible by the numbers from 1 (number), 1 to 4 (number), 4, and 6 (number), 6, a large number of divisors comparatively. It is central to many systems of timekeeping, including the Gregorian calendar, Western calendar and time, units of time of day, and frequently appears in the world's major religions. Name Twelve is the largest number with a monosyllable, single-syllable name in English language, English. Early Germanic languages, Germanic numbers have been theorized to have been non-decimal: evidence includes the unusual phrasing of 11 (number), eleven and twelve, the long hundred, former use of "hundred" to refer to groups of 120 (number), 120, and the presence of glosses such as "tentywise" or "ten-count" in medieval texts showing that writers could not pres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
18 (number)
18 (eighteen) is the natural number following 17 (number), 17 and preceding 19 (number), 19. It is an even composite number. Mathematics 18 is a semiperfect number and an abundant number. It is a largely composite number, as it has 6 divisors and no smaller number has more than 6 divisors. There are 18 One-sided polyomino, one-sided pentominoes. In the classification of finite simple groups, there are 18 infinite families of groups. In science Chemistry * The 18-Electron rule, 18-electron rule is a rule of thumb in transition metal chemistry for characterising and predicting the stability of Metal complex#Metal complexes, metal complexes. In religion and literature * The Hebrew language, Hebrew word for "life" is (''Chai (symbol), chai''), which has a gematria, numerical value of 18. Consequently, the custom has arisen in Jewish circles to give donations and monetary gifts in multiples of 18 as an expression of blessing for long life. * In Judaism, in the Talmud; Pirkei Avot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
36 (number)
36 (thirty-six) is the natural number following 35 (number), 35 and preceding 37 (number), 37. In mathematics 36 is both the Square number, square of 6, six, and the eighth triangular number or the sum of the first eight non-zero positive integers, which makes 36 the first non-trivial square triangular number. Aside from being the smallest square triangular number other than 1, it is also the only triangular number (other than 1) whose square root is also a triangular number. 36 is also the eighth refactorable number, as it has exactly nine positive divisors, and 9 is one of them; in fact, it is the smallest positive integer with at least nine divisors, which leads 36 to be the 7th highly composite number. It is the sum of the fourth pair of Twin prime, twin-primes (17 (number), 17 + 19 (number), 19), and the 18th Harshad number in Base ten, decimal, as it is divisible by the sum of its digits (9). It is the smallest number n with exactly eight solutions (37 (number), 37, 57 (nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
945 (number)
900 (nine hundred) is the natural number following 899 and preceding 901. It is the square of 30 and the sum of Euler's totient function for the first 54 positive integers. In base 10, it is a Harshad number. It is also the first number to be the square of a sphenic number. In other fields 900 is also: * A telephone area code for "premium" telephone calls in the North American Numbering Plan (900 number) * In Greek number symbols, the sign Sampi ("ϡ", literally "like a pi") * A skateboarding trick in which the skateboarder spins two and a half times (360 degrees times 2.5 is 900) * A 900 series refers to three consecutive perfect games in bowling * Yoda's age in Star Wars Integers from 901 to 999 900s * 901 = 17 × 53, centered triangular number, happy number * 902 = 2 × 11 × 41, sphenic number, nontotient, Harshad number * 903 = 3 × 7 × 43, sphenic number, 42nd triangular number, Schröder–Hipparchus number, Mertens function(903) returns 0, little Schroeder n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
104 (number)
104 (one hundred ndfour) is the natural number following 103 and preceding 105. In mathematics 104 is a refactorable number and a primitive semiperfect number. The smallest known 4-regular matchstick graph has 104 edges and 52 vertices, where four unit line segments intersect at every vertex. The second largest sporadic group \mathbb has a McKay–Thompson series, representative of a principal modular function is T_(\tau), with constant term a(0) = 104: :j_(\tau) = T_(\tau)+104 = \frac + 104 + 4372q + 96256q^2 + \cdots The Tits group \mathbb , which is the only finite simple group to classify as either a ''non-strict'' group of Lie type or sporadic group, holds a minimal faithful complex representation in 104 dimensions. References * Wells, D. ''The Penguin Dictionary of Curious and Interesting Numbers ''The Penguin Dictionary of Curious and Interesting Numbers'' is a reference book for recreational mathematics and elementary number theory written by David ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
88 (number)
88 (eighty-eight) is the natural number following 87 (number), 87 and preceding 89 (number), 89. In mathematics 88 is: * a refactorable number. * a primitive semiperfect number. * an untouchable number. * a polygonal number, hexadecagonal number. * an Erdős–Woods number, since it is possible to find sequences of 88 consecutive integers such that each inner member shares a factor with either the first or the last member. * a palindromic number in bases 5 (3235), 10 (8810), 21 (4421), and 43 (2243). * a repdigit in bases 10, 21 and 43. * a 2-automorphic number. * the smallest positive integer with a Zeckendorf representation requiring 5 Fibonacci numbers. * a strobogrammatic number. * the largest number in English not containing the letter 'n' in its name, when using Long and short scales, short scale. 88 and 945 are the smallest coprime abundant numbers, since all numbers until 945 are multiples of 2, 945 has 3, 5 and 7 as divisors, and 88 is the first abundant number that does ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mersenne Prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If is a composite number then so is . Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form for some prime . The exponents which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... . Numbers of the form without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that should be prime. The smallest composite Mersenne number with prime exponent ''n'' is . Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Eule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
