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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] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
289 (number)
289 is the natural number following 288 and preceding 290. In mathematics *289 is an odd composite number with only one prime factor. *289 is the 9th Friedman number A Friedman number is an integer, which represented in a given numeral system, is the result of a non-trivial expression using all its own digits in combination with any of the four basic arithmetic operators (+, −, ×, ÷), additive inverses, .... Friedman numbers are numbers that can be written by using its own digits the exact number of times they show up in the number. This one can be expressed as (8+9)2. *289 is a perfect square being equal to 172. It is also the 7th number to only have 3 factors because it is a square of a prime number. *289 is the sum of perfect cubes. It is the sum of 13+23+43+63. *289 is equivalent to the sum of the first 5 whole numbers to their respective powers. It is equal to 00+11+22+33+44. References External links {{Integers, 2 Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
291 (number)
291 is the natural number following 290 and preceding 292. In mathematics *291 is an odd composite number with two prime factors. *291 is a semiprime number meaning that it has 2 prime factors. *291 can be written as the sum of the nth prime A prime number (or a prime) is a natural number greater than 1 that is not a 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 ... plus n. It is the 52nd prime (239) plus 52. *291 is one of the positions of “c” in the tribonacci word abacabaab… defined by a->ab, b->ac, c->a. *291 is the sum of six 4th powers. It is the sum of 4⁴+2⁴+2⁴+1⁴+1⁴+1⁴. References {{Improve categories, date=October 2023 Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphenic Number
In number theory, a sphenic number (from , 'wedge') is a positive integer that is the product of three distinct prime numbers. Because there are infinitely many prime numbers, there are also infinitely many sphenic numbers. Definition A sphenic number is a product ''pqr'' where ''p'', ''q'', and ''r'' are three distinct prime numbers. In other words, the sphenic numbers are the square-free 3- almost primes. Examples The smallest sphenic number is 30 = 2 × 3 × 5, the product of the smallest three primes. The first few sphenic numbers are : 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, ... The largest known sphenic number at any time can be obtained by multiplying together the three largest known primes. Divisors All sphenic numbers have exactly eight divisors. If we express the sphenic number as n = p \cdot q \cdot r, where ''p'', ''q'', and ''r'' are distinct primes, then the set of divisors of ''n'' will be: :\left\. The converse does not hold. F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 nontotient if there is no integer ''x'' that has exactly ''n'' coprimes below it. All odd numbers are nontotients, except 1, since it has the solutions ''x'' = 1 and ''x'' = 2. The first few even nontotients are this sequence: : 14, 26, 34, 38, 50, 62, 68, 74, 76, 86, 90, 94, 98, 114, 118, 122, 124, 134, 142, 146, 152, 154, 158, 170, 174, 182, 186, 188, 194, 202, 206, 214, 218, 230, 234, 236, 242, 244, 246, 248, 254, 258, 266, 274, 278, 284, 286, 290, 298, ... The least value of ''k'' such that the totient of ''k'' is ''n'' are (0 if no such ''k'' exists) are this sequence: :1, 3, 0, 5, 0, 7, 0, 15, 0, 11, 0, 13, 0, 0, 0, 17, 0, 19, 0, 25, 0, 23, 0, 35, 0, 0, 0, 29, 0, 31, 0, 51, 0, 0, 0, 37, 0, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noncototient
In number theory, a noncototient is a positive integer that cannot be expressed as the difference between a positive integer and the number of coprime integers below it. That is, , where stands for 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 ot ..., has no solution for . The '' cototient'' of is defined as , so a noncototient is a number that is never a cototient. It is conjectured that all noncototients are even. This follows from a modified form of the slightly stronger version of the Goldbach conjecture: if the even number can be represented as a sum of two distinct primes and , then \begin pq - \varphi(pq) &= pq - (p-1)(q-1) \\ &= p + q - 1 \\ &= n - 1. \end It is expected that every even number larger than 6 is a sum of two disti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Mian–Chowla Sequence
In mathematics, the Mian–Chowla sequence is an integer sequence defined recursively in the following way. The sequence starts with :a_1 = 1. Then for n>1, a_n is the smallest integer such that every pairwise sum :a_i + a_j is distinct, for all i and j less than or equal to n. Properties Initially, with a_1, there is only one pairwise sum, 1 + 1 = 2. The next term in the sequence, a_2, is 2 since the pairwise sums then are 2, 3 and 4, i.e., they are distinct. Then, a_3 can't be 3 because there would be the non-distinct pairwise sums 1 + 3 = 2 + 2 = 4. We find then that a_3 = 4, with the pairwise sums being 2, 3, 4, 5, 6 and 8. The sequence thus begins : 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252 Year 252 ( CCLII) was a leap year starting on Thursday of the Julian calendar. At the time, it was known as the Year of the Consulship of Trebonianus and Volusianus (or, less frequently, year 1005 ''Ab urbe condita''). The denomination 252 for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
290 Theorem
In mathematics, the 15 theorem or Conway–Schneeberger Fifteen Theorem, proved by John H. Conway and W. A. Schneeberger in 1993, states that if a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers. The proof was complicated, and was never published. Manjul Bhargava found a much simpler proof which was published in 2000. Bhargava used the occasion of his receiving the 2005 SASTRA Ramanujan Prize to announce that he and Jonathan P. Hanke had cracked Conway's conjecture that a similar theorem holds for integral quadratic forms, with the constant 15 replaced by 290. The proof has since appeared in preprint form. Details Suppose Q_ is a symmetric matrix with real entries. For any vector x with integer components, define :Q(x) = x^t Q x = \sum_ x_i Q_ x_j This function is called a quadratic form. We say Q is positive definite if Q(x) > 0 whenever x \ne 0. If Q(x) is always an integer, we cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
