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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.Zachariou+Zachariou (1972) 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 |
Perfect Number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. The sum of divisors of a number, excluding the number itself, 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 including itself; in symbols, \sigma_1(n)=2n where \sigma_1 is the sum-of-divisors function. For instance, 28 is perfect as 1 + 2 + 4 + 7 + 14 = 28. This definition is ancient, appearing as early as Euclid's ''Elements'' (VII.22) where it is called (''perfect'', ''ideal'', or ''complete number''). Euclid also proved a formation rule (IX.36) whereby q(q+1)/2 is an even perfect number whenever q is a prime of the form 2^p-1 for positive integer p—what is now called a Mersenne prime. Two millenni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
40 (number)
40 (forty) is the natural number following 39 and preceding 41. Though the word is related to "four" (4), the spelling "forty" replaced "fourty" in the course of the 17th century and is now the standard form. In mathematics *Forty is a composite number, a refactorable number, an octagonal number, and—as the sum of the first four pentagonal numbers: 1 + 5 + 12 + 22 =40—it is a pentagonal pyramidal number. Adding up some subsets of its divisors (e.g., 1, 4, 5, 10, and 20) gives 40; hence, 40 is a semiperfect number. *Given 40, the Mertens function returns 0. 40 is the smallest number with exactly nine solutions to the equation Euler's totient function \varphi (x)=n. *Forty is the number of -queens problem solutions for n=7. *Forty is a repdigit in ternary (1111, ''i.e.'', 3^ + 3^ + 3^ + 3^, or, in other words, \frac ) and a Harshad number in decimal. In science *The atomic number of zirconium. *Negative forty is the unique temperature at which the Fahrenheit and C ... [...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 few 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 any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
12 (number)
12 (twelve) is the natural number following 11 and preceding 13. Twelve is a superior highly composite number, divisible by 2, 3, 4, and 6. It is the number of years required for an orbital period of Jupiter. It is central to many systems of timekeeping, including the Western calendar and units of time of day and frequently appears in the world's major religions. Name Twelve is the largest number with a single-syllable name in English. Early Germanic numbers have been theorized to have been non- decimal: evidence includes the unusual phrasing of eleven and twelve, the former use of "hundred" to refer to groups of 120, and the presence of glosses such as "tentywise" or "ten-count" in medieval texts showing that writers could not presume their readers would normally understand them that way. Such uses gradually disappeared with the introduction of Arabic numerals during the 12th-century Renaissance. Derived from Old English, ' and ' are first attested in the 10t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
18 (number)
18 (eighteen) is the natural number following 17 and preceding 19. In mathematics * Eighteen is a composite number, its divisors being 1, 2, 3, 6 and 9. Three of these divisors (3, 6 and 9) add up to 18, hence 18 is a semiperfect number. Eighteen is the first inverted square-prime of the form ''p''·''q''2. * In base ten, it is a Harshad number. * It is an abundant number, as the sum of its proper divisors is greater than itself (1+2+3+6+9 = 21). It is known to be a solitary number, despite not being coprime to this sum. * It is the number of one-sided pentominoes. * It is the only number where the sum of its written digits in base 10 (1+8 = 9) is equal to half of itself (18/2 = 9). * It is a Fine number. In science Chemistry * Eighteen is the atomic number of argon. * Group 18 of the periodic table is called the noble gases. * The 18-electron rule is a rule of thumb in transition metal chemistry for characterising and predicting the stability of metal com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
24 (number)
24 (twenty-four) is the natural number following 23 and preceding 25. The SI prefix for 1024 is yotta (Y), and for 10−24 (i.e., the reciprocal of 1024) yocto (y). These numbers are the largest and smallest number to receive an SI prefix to date. In mathematics 24 is an even composite number, with 2 and 3 as its distinct prime factors. It is the first number of the form 2''q'', where ''q'' is an odd prime. It is the smallest number with exactly eight positive divisors: 1, 2, 3, 4, 6, 8, 12, and 24; thus, it is a highly composite number, having more divisors than any smaller number. Furthermore, it is an abundant number, since the sum of its proper divisors ( 36) is greater than itself, as well as a superabundant number. In number theory and algebra *24 is the smallest 5-hemiperfect number, as it has a half-integer abundancy index: *:1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 = × 24 *24 is a semiperfect number, since adding up all the proper diviso ... [...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, 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, which is the second sphenic number. 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. * The measure of the central angle and exterior angle of a dodecagon, which is the petrie polygon of the 24-cell. * The number of sides of a triacontagon, which in turn is the petrie polygon of the 120-cel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
36 (number)
36 (thirty-six) is the natural number following 35 and preceding 37. In mathematics 36 is both the square of six and a triangular number, making it a square triangular number. It is the smallest square triangular number other than one, and it is also the only triangular number other than one whose square root is also a triangular number. It is also a Harshad number. It is the smallest number ''n'' with exactly eight solutions to the equation \phi(x)=n. It is the smallest number with exactly nine divisors, leading 36 to be a highly composite number. Adding up some subsets of its divisors (e.g., 6, 12, and 18) gives 36; hence, it is a semiperfect number. This number is the sum of the cubes of the first three positive integers and also the product of the squares of the first three positive integers. 36 is the number of degrees in the interior angle of each tip of a regular pentagram. The thirty-six officers problem is a mathematical puzzle with no solution. The number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abundant Number
In number theory, an abundant number or excessive number is a number 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 A number ''n'' for which the ''sum'' ''of'' ''divisors'' ''σ''(''n'') > 2''n'', or, equivalently, the sum of proper divisors (or aliquot sum) ''s''(''n'') > ''n''. Abundance is the value ''σ''(''n'') − ''2n'' (or ''s''(''n'') − ''n''). 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 = 12. Prop ... [...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 primitive semiperfect number and a composite number, with its divisors being 1, 2, 4, 8, 13, 26, 52 and 104. As it has 8 divisors total, and 8 is one of those divisors, 104 is a refactorable number. The distinct prime factors of 104 add up to 15, and so do the ones of 105, hence the two numbers form a Ruth-Aaron pair under the first definition. In regular geometry, 104 is the smallest number of unit line segments that can exist in a plane with four of them touching at every vertex.A figure made up of a row of 4 adjacent congruent rectangles is divided into 104 regions upon drawing diagonals of all possible rectangles φ(104) = φ(σ(104)). In science *The atomic number of rutherfordium. *Number of degrees Fahrenheit corresponding to 40 Celsius. In other fields 104 is also: *The number of Corinthian columns in the Temple of Olympian Zeus, the largest temple ever built in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
88 (number)
88 (eighty-eight) is the natural number following 87 and preceding 89. In mathematics 88 is: * a refactorable number. * a primitive semiperfect number. * an untouchable number. * a 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 short scale. 88 and 945 are the smallest coprime abundant numbers. In science and technology *The atomic number of the element radium. * The number of constellations in the sky as defined by the International Astronomical Union. * Messier object M ... [...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" phone calls in the North American Numbering Plan * 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, triangular number, Schröder–Hipparchus number, Mertens function (903) returns 0, little Schroeder number * 904 = 23 × 113 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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