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Untouchable Number
In mathematics, an untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer. That is, these numbers are not in the image of the aliquot sum function. Their study goes back at least to Abu Mansur al-Baghdadi (circa 1000 AD), who observed that both 2 and 5 are untouchable. Examples * The number 4 is not untouchable, as it is equal to the sum of the proper divisors of 9: 1 + 3 = 4. * The number 5 is untouchable, as it is not the sum of the proper divisors of any positive integer: 5 = 1 + 4 is the only way to write 5 as the sum of distinct positive integers including 1, but if 4 divides a number, 2 does also, so 1 + 4 cannot be the sum of all of any number's proper divisors (since the list of factors would have to contain both 4 and 2). * The number 6 is not untouchable, as it is equal to the sum of the proper divisors of 6 itself: 1 + 2 + 3&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
206 (number)
206 (two hundred ndsix) is the natural number following 205 and preceding 207. In mathematics 206 is both a nontotient and a noncototient. 206 is an untouchable number. It is the lowest positive integer (when written in English as "two hundred and six") to employ all of the vowels once only, not including Y. The other numbers sharing this property are 230, 250, 260, 602, 640, 5000, 8000, 9000, 26,000, 80,000 and 90,000. 206 and 207 form the second pair of consecutive numbers (after 14 and 15) whose sums of divisors are equal. There are exactly 206 different linear forests on five labeled nodes, and exactly 206 regular semigroup In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of semigroup ...s of order four up to isomorphism and anti-isomorphism. References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
306 (number)
306 is the natural number following 305 and preceding 307. In mathematics *306 is an even composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime numb ... with three prime factors. *306 is the sum of consecutive primes 71+73+79+83. *306 is the 17th oblong number meaning that it is equal to 17*18. *306 is an untouchable number meaning that it cannot be equal to the sum of proper factors in any number. *There are 306 triangular numbers with 5 digits. References Integers {{Num-stub ... [...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] |
292 (number)
292 is the natural number following 291 and preceding 293. In mathematics *292 is an even composite number with two prime factors. *292 is a noncototient number meaning that phi(x) cannot result in 292. *292 is an untouchable number meaning that the proper divisors of any number do not add up to 292. *292 is a repdigit in base 8 with it being 444. *In the simplified continued fraction A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ... for pi, 292 is the 5th number. References {{Improve categories, date=November 2023 Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
290 (number)
290 (two hundred ndninety) is the natural number following 289 and preceding 291. In mathematics The product of three primes, 290 is a sphenic number, and the sum of four consecutive primes (67 + 71 + 73 + 79). The sum of the squares of the divisors of 17 is 290. Not only is it 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 ... and a noncototient, it is also an untouchable number. 290 is the 16th member of the Mian–Chowla sequence; it can not be obtained as the sum of any two previous terms in the sequence. See also the Bhargava–Hanke 290 theorem. References {{DEFAULTSORT:290 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
288 (number)
288 (two hundred ndeighty-eight) is the natural number following 287 and preceding 289. Because 288 = 2 · 12 · 12, it may also be called "two gross" or "two dozen dozen". In mathematics Factorization properties Because its prime factorization 288 = 2^5\cdot 3^2 contains only the first two prime numbers 2 and 3, 288 is a 3-smooth number. This factorization also makes it a highly powerful number, a number with a record-setting value of the product of the exponents in its factorization. Among the highly abundant numbers, numbers with record-setting sums of divisors, it is one of only 13 such numbers with an odd divisor sum. Both 288 and are powerful numbers, numbers in which all exponents of the prime factorization are larger than one. This property is closely connected to being highly abundant with an odd divisor sum: all sufficiently large highly abundant numbers have an odd prime factor with exponent one, causing their divisor sum to be even. 288 and 289 form only the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
276 (number)
276 (two hundred ndseventy-six) is the natural number following 275 and preceding 277. In mathematics 276 is the sum of 3 consecutive fifth powers (276 = 15 + 25 + 35). As a figurate number it is the 23rd triangular number, a hexagonal number, and a centered pentagonal number, the third number after 1 and 6 to have this combination of properties. 276 is the first triangular number that can be arrived at in three ways by adding pairs of triangular numbers together. This sequence, dubbed 'Triple Triangle-Pair Numbers' is the sequence of integers: 276, 406, 666, ... 276 is the size of the largest set of equiangular lines in 23 dimensions. The maximal set of such lines, derived from the Leech lattice, provides the highest dimension in which the "Gerzon bound" of \binom is known to be attained; its symmetry group is the third Conway group, Co3. 276 is the smallest number for which it is not known if the corresponding aliquot sequence either terminates or ends in a repeating cyc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
268 (number)
268 (two hundred ndsixty-eight) is the natural number following 267 and preceding 269. In mathematics *268 is an even composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime numb ... with two prime factors, but one of the prime factors is repeated: 268 = 67*2*2. *268 is the smallest number whose product of digits is 6 times the sum of its digits. *268 is untouchable which means that it is not the sum of the proper divisors of any number *268 is the sum of the consecutive primes 131 and 137. References {{Improve categories, date=October 2023 Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
262 (number)
262 (two hundred ndsixty-two) is a natural number preceded by the number 261 and followed by 263. It has the prime factorization 2·131. Mathematical properties There are four divisors of this number, the divisors being , , , and 262 itself, which makes it a semiprime. It is the sixth meandric number, and the ninth open meandric number. As it cannot be divided into the sum of the proper divisors of any number, it is the 17th untouchable number. As it eventually reaches 1 when replaced by the sum of the square of each digit, it is the 40th 10- happy number. As 262 is 262 backwards, it is a palindromic number A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16361) that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis. The term ''palin .... References {{Integers, 2 Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
248 (number)
248 (two hundred ndforty-eight) is the natural number following 247 and preceding 249. Additionally, 248 is: *a nontotient. *a refactorable number. *an untouchable number. *palindromic in bases 13 (16113), 30 (8830), 61 (4461) and 123 (22123). *a Harshad number in bases 3, 4, 6, 7, 9, 11, 13 (and 18 other bases). *part of the 43-aliquot tree. The aliquot sequence starting at 248 is: 248, 232, 218, 112, 136, 134, 70, 74, 40, 50, 43, 1, 0. The exceptional Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ... E8 has dimension 248. References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
246 (number)
246 (two hundred ndforty-six) is the natural number following 245 and preceding 247. Additionally, 246 is: *an untouchable number. *palindromic in bases 5 (14415), 9 (3039), 40 (6640), 81 (3381), 122 (22122) and 245 (11245). *a Harshad number in bases 2, 3, 6, 7, 9, 11 (and 15 other bases). *the smallest number N for which it is known that there is an infinite number of prime gaps no larger than N. Also: *The aliquot sequence starting at 246 is: 246, 258, 270, 450, 759, 393, 135, 105, 87, 33, 15, 9, 4, 3, 1, 0. *There are exactly 246 different rooted plane trees with eight nodes, and 246 different necklaces A necklace is an article of jewellery that is worn around the neck. Necklaces may have been one of the earliest types of adornment worn by humans. They often serve Ceremony, ceremonial, Religion, religious, magic (illusion), magical, or Funerar ... with seven black and seven white beads. References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
