"a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once. Integer sequence 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:"
"number. In ""A Mathematician's Apology"", G. H. Hardy wrote: The sequence of base 10 narcissistic numbers starts: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, ... The sequence of base 8 narcissistic numbers starts: 0, 1, 2, 3, 4, 5, 6, 7, 24, 64, 134, 205, 463, 660, 661, ... (sequence and in OEIS) The sequence of base 12 narcissistic numbers starts: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ᘔ, Ɛ, 25, ᘔ5, 577, 668, ᘔ83, ... The sequence of base 16 narcissistic numbers starts: 0, 1, 2,"
"Golomb sequence In mathematics, the Golomb sequence, named after Solomon W. Golomb (but also called Silverman's sequence), is a non-decreasing integer sequence where ""a"" is the number of times that ""n"" occurs in the sequence, starting with ""a"" = 1, and with the property that for ""n"" > 1 each ""a"" is the unique integer which makes it possible to satisfy the condition. For example, ""a"" = 1 says that 1 only occurs once in the sequence, so ""a"" cannot be 1 too, but it can be, and therefore must be, 2. The first few values are ""a"" = 1"
"Golomb sequence In mathematics, the Golomb sequence, named after Solomon W. Golomb (but also called Silverman's sequence), is a non-decreasing integer sequence where ""a"" is the number of times that ""n"" occurs in the sequence, starting with ""a"" = 1, and with the property that for ""n"" > 1 each ""a"" is the unique integer which makes it possible to satisfy the condition. For example, ""a"" = 1 says that 1 only occurs once in the sequence, so ""a"" cannot be 1 too, but it can be, and therefore must be, 2. The first few values are ""a"" = 1"
"Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is, and for . One has . In some books, and particularly in old ones, , the ""0"" is omitted, and the Fibonacci sequence starts with . The beginning of the Fibonacci sequence is thus: Fibonacci numbers are strongly related to the golden ratio: Binet's formula expresses the th Fibonacci number in terms of and the golden ratio, and imply that the ratio of two consecutive"
"formula_3. A more formal definition uses a recurrence relation. Define the numbers formula_8 (with ""k"" ≥ ""n"" ≥ 0) by Then the transformed sequence is defined by formula_11. In the case ""a"" = 1, ""a"" = 0 (""n"" > 0), the resulting triangle is called the Seidel–Entringer–Arnold Triangle and the numbers formula_8 are called Entringer numbers . In this case the numbers in the transformed sequence ""b"" are called the Euler up/down numbers. This is sequence on the On-Line Encyclopedia of Integer Sequences. These enumerate the number of alternating permutations on ""n"" letters and are related to the Euler numbers"
"sequence with the three leading zeros deleted is obtained as the sequence of coefficients of the power-series expansion of This sequence has also been called Alcuin's sequence by some authors. Alcuin's sequence In mathematics, Alcuin's sequence, named after Alcuin of York, is the sequence of coefficients of the power-series expansion of: The sequence begins with these integers: The ""n""th term is the number of triangles with integer sides and perimeter ""n"". It is also the number of triangles with ""distinct"" integer sides and perimeter ""n"" + 6, i.e. number of triples (""a"", ""b"", ""c"") such that 1 ≤ ""a"" <"
"about real numbers that are rational numbers, and is expressed symbolically by writing The most familiar numbers are the natural numbers (sometimes called whole numbers or counting numbers): 1, 2, 3, and so on. Traditionally, the sequence of natural numbers started with 1 (0 was not even considered a number for the Ancient Greeks.) However, in the 19th century, set theorists and other mathematicians started including 0 (cardinality of the empty set, i.e. 0 elements, where 0 is thus the smallest cardinal number) in the set of natural numbers. Today, different mathematicians use the term to describe both sets, including"
"Hilbert number In number theory, a branch of mathematics, a Hilbert number is a positive integer of the form (). The Hilbert numbers were named after David Hilbert. The sequence of Hilbert numbers begins 1, 5, 9, 13, 17, ... ). A Hilbert prime is a Hilbert number that is not divisible by a smaller Hilbert number (other than 1). The sequence of Hilbert primes begins A Hilbert prime is not necessarily a prime number; for example, 21 is a composite number since . However, 21 a Hilbert prime since neither 3 nor 7 (the only factors of 21 other"
"Alcuin's sequence In mathematics, Alcuin's sequence, named after Alcuin of York, is the sequence of coefficients of the power-series expansion of: The sequence begins with these integers: The ""n""th term is the number of triangles with integer sides and perimeter ""n"". It is also the number of triangles with ""distinct"" integer sides and perimeter ""n"" + 6, i.e. number of triples (""a"", ""b"", ""c"") such that 1 ≤ ""a"" < ""b"" < ""c"" < ""a"" + ""b"", ""a"" + ""b"" + ""c"" = ""n"" + 6. If one deletes the three leading zeros, then it is the number of ways"
"moves a draw. Thue–Morse sequence In mathematics, the Thue–Morse sequence, or Prouhet–Thue–Morse sequence, is the binary sequence (an infinite sequence of 0s and 1s) obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far. The first few steps of this procedure yield the strings 0 then 01, 0110, 01101001, 0110100110010110, and so on, which are prefixes of the Thue–Morse sequence. The full sequence begins: There are several equivalent ways of defining the Thue–Morse sequence. To compute the ""n""th element ""t"", write the number ""n"" in binary. If the number of ones in this"
"to /5. Repeating this process we get an infinite sequence of numbers , 2, /5, ... which converge to 3. These numbers are called the Lagrange numbers, and are named after Joseph Louis Lagrange. The ""n""th Lagrange number ""L"" is given by where ""m"" is the ""n""th Markov number, that is the ""n""th smallest integer ""m"" such that the equation has a solution in positive integers ""x"" and ""y"". Lagrange number In mathematics, the Lagrange numbers are a sequence of numbers that appear in bounds relating to the approximation of irrational numbers by rational numbers. They are linked to Hurwitz's"
"than 1 and itself) are Hilbert numbers. It follows from multiplication modulo 4 that a Hilbert prime is either a prime number of the form (called a Pythagorean prime), or a semiprime of the form . Hilbert number In number theory, a branch of mathematics, a Hilbert number is a positive integer of the form (). The Hilbert numbers were named after David Hilbert. The sequence of Hilbert numbers begins 1, 5, 9, 13, 17, ... ). A Hilbert prime is a Hilbert number that is not divisible by a smaller Hilbert number (other than 1). The sequence of Hilbert"
"sequence has been found. For instance, starting with ""n"" = 12, one gets the sequence 12, 6, 3, 10, 5, 16, 8, 4, 2, 1. ""n"" = 19, for example, takes longer to reach 1: 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. The sequence for ""n"" = 27, listed and graphed below, takes 111 steps (41 steps through odd numbers, in large font), climbing to 9232 before descending to 1. Numbers with a total stopping time longer than that of any smaller starting value form a"
"Sylvester's sequence In number theory, Sylvester's sequence is an integer sequence in which each member of the sequence is the product of the previous members, plus one. The first few terms of the sequence are: Sylvester's sequence is named after James Joseph Sylvester, who first investigated it in 1880. Its values grow doubly exponentially, and the sum of its reciprocals forms a series of unit fractions that converges to 1 more rapidly than any other series of unit fractions with the same number of terms. The recurrence by which it is defined allows the numbers in the sequence to be"
"Farey sequence In mathematics, the Farey sequence of order ""n"" is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which when in lowest terms have denominators less than or equal to ""n"", arranged in order of increasing size. With the restricted definition, each Farey sequence starts with the value 0, denoted by the fraction ⁄, and ends with the value 1, denoted by the fraction ⁄ (although some authors omit these terms). A Farey sequence is sometimes called a Farey series, which is not strictly correct, because the terms are not summed. The"
"starts as follows: Ilan Vardi has called this sequence, starting with ""d"" = 3, the Conway sequence . (for ""d"" = 2, see ) The sequence grows indefinitely. In fact, any variant defined by starting with a different integer seed number will (eventually) also grow indefinitely, except for the degenerate sequence: 22, 22, 22, 22, … No digits other than 1, 2, and 3 appear in the sequence, unless the seed number contains such a digit or a run of more than three of the same digit. Conway's cosmological theorem asserts that every sequence eventually splits (""decays"") into a sequence"
"Pillai sequence The Pillai sequence is the sequence of integers that have a record number of terms in their greedy representations as sums of prime numbers (and one). It is named after Subbayya Sivasankaranarayana Pillai, who first defined it in 1930. It would follow from Goldbach's conjecture that every integer greater than one can be represented as a sum of at most three prime numbers. However, finding such a representation could involve solving instances of the subset sum problem, which is computationally difficult. Instead, Pillai considered the following simpler greedy algorithm for finding a representation of formula_1 as a sum"
"Ulam number An Ulam number is a member of an integer sequence devised by and named after Stanislaw Ulam, who introduced it in 1964. The standard Ulam sequence (the (1, 2)-Ulam sequence) starts with ""U"" = 1 and ""U"" = 2. Then for ""n"" > 2, ""U"" is defined to be the smallest integer that is the sum of two distinct earlier terms in exactly one way and larger than all earlier terms. As a consequence of the definition, 3 is an Ulam number (1+2); and 4 is an Ulam number (1+3). (Here 2+2 is not a second representation of"
"the relative order of the entries. By convention, in any naming scheme the unique permutations of length 0 (the permutation of the empty set) and 1 (the permutation consisting of a single entry 1) are taken to be alternating. The determination of the number ""A"" of alternating permutations of the set {1, ..., ""n""} is called ""André's problem"". The numbers ""A"" are variously known as ""Euler numbers"", ""zigzag numbers"", ""up/down numbers"", or by some combinations of these names. The name Euler numbers in particular is sometimes used for a closely related sequence. The first few values of ""A"" are 1,"
"3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 156, 173, 208, 248, 285, 4A5, 5B0, 5B1, 60B, 64B, ... The sequence of base 3 narcissistic numbers starts: 0, 1, 2, 12, 22, 122 The sequence of base 4 narcissistic numbers starts: 0, 1, 2, 3, 130, 131, 203, 223, 313, 332, 1103, 3303 (sequence and in OEIS) In base 2, the only narcissistic numbers are 0 and 1. The number of narcissistic numbers in a given base is finite, since the maximum possible sum of the ""k""th powers of a ""k"" digit number in base"
"Aliquot sequence In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0. The aliquot sequence starting with a positive integer ""k"" can be defined formally in terms of the sum-of-divisors function σ or the aliquot sum function ""s"" in the following way: and ""s""(0) is undefined. For example, the aliquot sequence of 10 is 10, 8, 7, 1, 0 because: Many aliquot sequences terminate"
"Thue–Morse sequence In mathematics, the Thue–Morse sequence, or Prouhet–Thue–Morse sequence, is the binary sequence (an infinite sequence of 0s and 1s) obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far. The first few steps of this procedure yield the strings 0 then 01, 0110, 01101001, 0110100110010110, and so on, which are prefixes of the Thue–Morse sequence. The full sequence begins: There are several equivalent ways of defining the Thue–Morse sequence. To compute the ""n""th element ""t"", write the number ""n"" in binary. If the number of ones in this binary expansion is"
"Tetradic number A tetradic number, also known as a four-way number, is a number that remains the same when flipped back to front, flipped front to back, mirrored up-down, or flipped up-down. The only numbers that remain the same which turned up-side-down or mirrored are 0, 1, and 8, so a tetradic number is a palindromic number containing only 0, 1, and 8 as digits. The first few tetradic numbers are 1, 8, 11, 88, 101, 111, 181, 808, 818, ... (OEIS A006072). Tetradic numbers are also known as four-way numbers due to the fact that they have four-way symmetry"
"puzzle ""What is the next number in the sequence 1, 11, 21, 1211, 111221?"" is sometimes referred to as the Cuckoo's Egg, from a description of Morris in Clifford Stoll's book ""The Cuckoo's Egg"". There are many possible variations on the rule used to generate the look-and-say sequence. For example, to form the ""pea pattern"" one reads the previous term and counts all instances of each digit, listed in order of their first appearance, not just those occurring in a consecutive block. Thus, beginning with the seed 1, the pea pattern proceeds 1, 11 (""one 1""), 21 (""two 1s""), 1211"
"Lollipop sequence numbering Lollipop sequence numbering is a numbering scheme used in routing protocols. In this numbering scheme, sequence numbers start at a negative value, increase until they reach zero, then cycle through a finite set of positive numbers indefinitely. When a system is rebooted, the sequence is restarted from a negative number again. This allows recently rebooted systems to be distinguished from systems which have simply looped around their numbering space. This path can be visualized as a line with a circle at the end; hence a lollipop. Lollipop sequence numbering was originally believed to resolve the ambiguity problem"
"Rudin–Shapiro sequence In mathematics, the Rudin–Shapiro sequence, also known as the Golay–Rudin–Shapiro sequence, is an infinite automatic sequence named after Marcel Golay, Walter Rudin, and Harold S. Shapiro, who independently investigated its properties. Each term of the Rudin–Shapiro sequence is either +1 or −1. The ""n"" term of the sequence, ""b"", is defined by the rules: where formula_3 are the digits in the binary expansion of ""n"". Thus ""a"" counts the number of (possibly overlapping) occurrences of the sub-string 11 in the binary expansion of ""n"", and ""b"" is +1 if ""a"" is even and −1 if ""a"" is odd."
"sequence continues 1, 1, 2, 3, 5, 10, 28, 154... . ""a"" is the first term of this sequence that is not an integer. 43 is a Heegner number. 43 is the largest prime which dives the order of the Janko group J. 43 is a repdigit in base 6 (111). 43 is the largest natural number that is not a (original) McNugget number. 43 is the smallest prime number expressible as the sum of 2, 3, 4, or 5 different primes: When taking the first six terms of the Taylor series for computing e, one obtains which is also"
"them; that is, infinite arithmetic progressions form a Helly family. However, the intersection of infinitely many infinite arithmetic progressions might be a single number rather than itself being an infinite progression. If then Arithmetic progression In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. Difference here means the second minus the first. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with ""common difference"" of 2. If the initial term of an arithmetic progression is formula_1 and"
"Euclid–Mullin sequence The Euclid–Mullin sequence is an infinite sequence of distinct prime numbers, in which each element is the least prime factor of one plus the product of all earlier elements. They are named after the ancient Greek mathematician Euclid, because their definition relies on an idea in Euclid's proof that there are infinitely many primes, and after Albert A. Mullin, who asked about the sequence in 1963. The first 51 elements of the sequence are These are the only known elements . Finding the next one requires finding the least prime factor of a 335-digit number (which is known"
"Juggler sequence In recreational mathematics a juggler sequence is an integer sequence that starts with a positive integer ""a"", with each subsequent term in the sequence defined by the recurrence relation: 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"" = 3 is If a juggler sequence reaches 1, then all subsequent terms are equal to 1. It is conjectured that all juggler sequences eventually reach 1. This"
"Sign sequence In mathematics, a sign sequence, or ±1–sequence or bipolar sequence, is a sequence of numbers, each of which is either 1 or −1. One example is the sequence (1, −1, 1, −1 ...). Such sequences are commonly studied in discrepancy theory. Around 1932, mathematician Paul Erdős conjectured that for any infinite ±1-sequence formula_1 and any integer ""C"", there exist integers ""k"" and ""d"" such that The Erdős Discrepancy Problem asks for a proof or disproof of this conjecture. In February 2014, Alexei Lisitsa and Boris Konev of the University of Liverpool showed that every sequence of 1161 or"
"element is the sum of the previous elements (with the exception of the first elements in the sequence). The usual Fibonacci numbers are a Fibonacci sequence of order 2. The cases and have been thoroughly investigated. The number of compositions of nonnegative integers into parts that are at most is a Fibonacci sequence of order . The sequence of the number of strings of 0s and 1s of length that contain at most consecutive 0s is also a Fibonacci sequence of order . These sequences, their limiting ratios, and the limit of these limiting ratios, were investigated by Mark Barr"