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Integer Sequences
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description. The sequence 0, 3, 8, 15, ... is formed according to the formula ''n''2 − 1 for the ''n''th term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, even though we do not have a formula for the ''n''th perfect number. Examples Integer sequences that have their own name include: *Abundant numbers *Baum–Sweet sequence *Bell numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goteborg Ciag Fibonacciego
Gothenburg (; abbreviated Gbg; sv, Göteborg ) is the second-largest city in Sweden, fifth-largest in the Nordic countries, and capital of the Västra Götaland County. It is situated by the Kattegat, on the west coast of Sweden, and has a population of approximately 590,000 in the city proper and about 1.1 million inhabitants in the metropolitan area. Gothenburg was founded as a heavily fortified, primarily Dutch, trading colony, by royal charter in 1621 by King Gustavus Adolphus. In addition to the generous privileges (e.g. tax relaxation) given to his Dutch allies from the ongoing Thirty Years' War, the king also attracted significant numbers of his German and Scottish allies to populate his only town on the western coast. At a key strategic location at the mouth of the Göta älv, where Scandinavia's largest drainage basin enters the sea, the Port of Gothenburg is now the largest port in the Nordic countries. Gothenburg is home to many students, as the city includes the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Word
A Fibonacci word is a specific sequence of binary digits (or symbols from any two-letter alphabet). The Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers are formed by repeated addition. It is a paradigmatic example of a Sturmian word and specifically, a morphic word. The name "Fibonacci word" has also been used to refer to the members of a formal language ''L'' consisting of strings of zeros and ones with no two repeated ones. Any prefix of the specific Fibonacci word belongs to ''L'', but so do many other strings. ''L'' has a Fibonacci number of members of each possible length. Definition Let S_0 be "0" and S_1 be "01". Now S_n = S_S_ (the concatenation of the previous sequence and the one before that). The infinite Fibonacci word is the limit S_, that is, the (unique) infinite sequence that contains each S_n, for finite n, as a prefix. Enumerating items from the above definition produces: S_0 0 S_1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal number, cardinal numbers'', and numbers used for ordering are called ''Ordinal number, ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports Number (sports), jersey numbers). Some definitions, including the standard ISO/IEC 80000, ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Motzkin Number
In mathematics, the th Motzkin number is the number of different ways of drawing non-intersecting chords between points on a circle (not necessarily touching every point by a chord). The Motzkin numbers are named after Theodore Motzkin and have diverse applications in geometry, combinatorics and number theory. The Motzkin numbers M_n for n = 0, 1, \dots form the sequence: : 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209, 593742784829, ... Examples The following figure shows the 9 ways to draw non-intersecting chords between 4 points on a circle (): : The following figure shows the 21 ways to draw non-intersecting chords between 5 points on a circle (): : Properties The Motzkin numbers satisfy the recurrence relations :M_=M_+\sum_^M_iM_=\fracM_+\fracM_. The Motzkin numbers can be express ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucas Number
The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences. The Lucas series has the same recursive relationship as the Fibonacci sequence, where each term is the sum of the two previous terms, but with different starting values. This produces a sequence where the ratios of successive terms approach the golden ratio, and in fact the terms themselves are roundings of integer powers of the golden ratio. The sequence also has a variety of relationships with the Fibonacci numbers, like the fact that adding any two Fibonacci numbers two terms apart in the Fibonacci sequence results in the Lucas number in between. The first few Lucas numbers are : 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349 .... Defini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucky Number
In number theory, a lucky number is a natural number in a set which is generated by a certain "sieve". This sieve is similar to the Sieve of Eratosthenes that generates the primes, but it eliminates numbers based on their position in the remaining set, instead of their value (or position in the initial set of natural numbers). The term was introduced in 1956 in a paper by Gardiner, Lazarus, Metropolis and Ulam. They suggest also calling its defining sieve, "the sieve of Josephus Flavius" because of its similarity with the counting-out game in the Josephus problem. Lucky numbers share some properties with primes, such as asymptotic behaviour according to the prime number theorem; also, a version of Goldbach's conjecture has been extended to them. There are infinitely many lucky numbers. Twin lucky numbers and twin primes also appear to occur with similar frequency. However, if ''L''''n'' denotes the ''n''-th lucky number, and ''p''''n'' the ''n''-th prime, then ''L''''n'' > ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kolakoski Sequence
In mathematics, the Kolakoski sequence, sometimes also known as the Oldenburger–Kolakoski sequence, is an infinite sequence of symbols that is the sequence of run lengths in its own run-length encoding. It is named after the recreational mathematician William Kolakoski (1944–97) who described it in 1965, but it was previously discussed by Rufus Oldenburger in 1939. Definition The initial terms of the Kolakoski sequence are: :1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,1,1,... Each symbol occurs in a "run" (a sequence of equal elements) of either one or two consecutive terms, and writing down the lengths of these runs gives exactly the same sequence: :1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,1,1,2,1,1,2,1,2,2,1,2,2,1,1,2,1,2,2,... :1, 2 , 2 ,1,1, 2 ,1, 2 , 2 ,1, 2 , 2 ,1,1, 2 ,1,1, 2 , 2 ,1, 2 ,1,1, 2 ,1, 2 , 2&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Juggler Sequence
In number theory, a juggler sequence is an integer sequence that starts with a positive integer ''a''0, with each subsequent term in the sequence defined by the recurrence relation: a_= \begin \left \lfloor a_k^ \right \rfloor, & \text a_k \text \\ \\ \left \lfloor a_k^ \right \rfloor, & \text a_k \text. \end Background Juggler sequences were publicised by American mathematician and author Clifford A. Pickover. The name is derived from the rising and falling nature of the sequences, like balls in the hands of a juggler. For example, the juggler sequence starting with ''a''0 = 3 is :a_1= \lfloor 3^\frac \rfloor = \lfloor 5.196\dots \rfloor = 5, :a_2= \lfloor 5^\frac \rfloor = \lfloor 11.180\dots \rfloor = 11, :a_3= \lfloor 11^\frac \rfloor = \lfloor 36.482\dots \rfloor = 36, :a_4= \lfloor 36^\frac \rfloor = \lfloor 6 \rfloor = 6, :a_5= \lfloor 6^\frac \rfloor = \lfloor 2.449\dots \rfloor = 2, :a_6= \lfloor 2^\frac \rfloor = \lfloor 1.414\dots \rfloor = 1. If a juggler ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperperfect Number
In mathematics, a ''k''-hyperperfect number is a natural number ''n'' for which the equality ''n'' = 1 + ''k''(''σ''(''n'') − ''n'' − 1) holds, where ''σ''(''n'') is the divisor function (i.e., the sum of all positive divisors of ''n''). A hyperperfect number is a ''k''-hyperperfect number for some integer ''k''. Hyperperfect numbers generalize perfect numbers, which are 1-hyperperfect. The first few numbers in the sequence of ''k''-hyperperfect numbers are 6, 21, 28, 301, 325, 496, 697, ... , with the corresponding values of ''k'' being 1, 2, 1, 6, 3, 1, 12, ... . The first few ''k''-hyperperfect numbers that are not perfect are 21, 301, 325, 697, 1333, ... . List of hyperperfect numbers The following table lists the first few ''k''-hyperperfect numbers for some values of ''k'', together with the sequence number in the On-Line Encyclopedia of Integer Sequences (OEIS) of the sequence of ''k''-hyperperfect numbers: It can be shown that if ''k'' > 1 is an odd intege ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Home Prime
In number theory, the home prime HP(''n'') of an integer ''n'' greater than 1 is the prime number obtained by repeatedly factoring the increasing concatenation of prime factors including repetitions. The ''m''th intermediate stage in the process of determining HP(''n'') is designated HPn(''m''). For instance, HP(10) = 773, as 10 factors as 2×5 yielding HP10(1) = 25, 25 factors as 5×5 yielding HP10(2) = HP25(1) = 55, 55 = 5×11 implies HP10(3) = HP25(2) = HP55(1) = 511, and 511 = 7×73 gives HP10(4) = HP25(3) = HP55(2) = HP511(1) = 773, a prime number. Some sources use the alternative notation HPn for the homeprime, leaving out parentheses. Investigations into home primes make up a minor side issue in number theory. Its questions have served as test fields for the implementation of efficient algorithms for factoring composite numbers, but the subject ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Highly Totient Number
A highly totient number k is an integer that has more solutions to the equation \phi(x) = k, where \phi is Euler's totient function, than any integer below it. The first few highly totient numbers are 1, 2, 4, 8, 12, 24, 48, 72, 144, 240, 432, 480, 576, 720, 1152, 1440 , with 1, 3, 4, 5, 6, 10, 11, 17, 21, 31, 34, 37, 38, 49, 54, and 72 totient solutions respectively. The sequence of highly totient numbers is a subset of the sequence of smallest number k with exactly n solutions to \phi(x) = k. The totient of a number x, with prime factorization x=\prod_i p_i^, is the product: :\phi(x)=\prod_i (p_i-1)p_i^. Thus, a highly totient number is a number that has more ways of being expressed as a product of this form than does any smaller number. The concept is somewhat analogous to that of highly composite numbers, and in the same way that 1 is the only odd highly composite number, it is also the only odd highly totient number (indeed, the only odd number to not be a nontotient ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Highly Composite Number
__FORCETOC__ A highly composite number is a positive integer with more divisors than any smaller positive integer has. The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller positive integer. The name can be somewhat misleading, as the first two highly composite numbers (1 and 2) are not actually composite numbers; however, all further terms are. The late mathematician Jean-Pierre Kahane has suggested that Plato must have known about highly composite numbers as he deliberately chose 5040 as the ideal number of citizens in a city as 5040 has more divisors than any numbers less than it. Ramanujan wrote and titled his paper on the subject in 1915. Examples The initial or smallest 38 highly composite numbers are listed in the table below . The number of divisors is given in the column labeled ''d''(''n''). Asterisks indicate superior highly composite numbers. The divisors of the first 15 highly composite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
