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OEIS | A Wisdom Archive on OEIS | | OEIS A selection of articles related to OEIS | |
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oeis, On-Line Encyclopedia of Integer Sequences, On-Line Encyclopedia of Integer Sequences - An abridged example of a typical OEIS entry, On-Line Encyclopedia of Integer Sequences - Conventions, On-Line Encyclopedia of Integer Sequences - Entry fields, On-Line Encyclopedia of Integer Sequences - Errors or problems in the OEIS, On-Line Encyclopedia of Integer Sequences - History, On-Line Encyclopedia of Integer Sequences - Non-integers, On-Line Encyclopedia of Integer Sequences - Searching the OEIS, On-Line Encyclopedia of Integer Sequences - Self-referentiality, On-Line Encyclopedia of Integer Sequences - Authors, On-Line Encyclopedia of Integer Sequences - Comments, On-Line Encyclopedia of Integer Sequences - Enter a sequence, On-Line Encyclopedia of Integer Sequences - Enter a sequence number, On-Line Encyclopedia of Integer Sequences - Enter a word, On-Line Encyclopedia of Integer Sequences - ID number, On-Line Encyclopedia of Integer Sequences - Keywords, On-Line Encyclopedia of Integer Sequences - Lexicographic ordering, On-Line Encyclopedia of Integer Sequences - Maple Mathematica and other programs, On-Line Encyclopedia of Integer Sequences - Name, On-Line Encyclopedia of Integer Sequences - Offset, On-Line Encyclopedia of Integer Sequences - Sequence, On-Line Encyclopedia of Integer Sequences - Signed, On-Line Encyclopedia of Integer Sequences - Special meaning of zero, On-Line Encyclopedia of Integer Sequences - URL
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| ARTICLES RELATED TO OEIS | | | |  OEIS: Encyclopedia II - Sudoku - VariantsAlthough the 9×9 grid with 3×3 regions is by far the most common, numerous variations abound: sample puzzles can be 4×4 grids with 2×2 regions; 5×5 grids with pentomino regions have been published under the name Logi-5; the World Puzzle Championship has previously featured a 6×6 grid with 2×3 regions and a 7×7 grid with six heptomino regions and a disjoint region; Daily SuDoku features new 4×4, 6×6, and simpler 9×9 grids every day as Daily SuDoku for Kids. [1] Even the 9×9 grid is not always standard, with Ebb regul ...
See also:Sudoku, Sudoku - Introduction, Sudoku - Rules and terminology, Sudoku - Solution methods, Sudoku - Scanning, Sudoku - Marking up, Sudoku - Analysis, Sudoku - Computer solutions, Sudoku - Difficulty ratings, Sudoku - Construction, Sudoku - Variants, Sudoku - Mathematics of Sudoku, Sudoku - History, Sudoku - Popularity in the media Read more here: » Sudoku: Encyclopedia II - Sudoku - Variants |
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| | | | | | OEIS: Encyclopedia II - Sudoku - Difficulty ratings Published puzzles often are ranked in terms of difficulty. Perhaps surprisingly, the number of givens has little or no bearing on a puzzle's difficulty. A puzzle with a minimum number of givens may be very easy to solve, and a puzzle with more than the average number of givens can still be extremely difficult to solve. It is based on the relevance and the positioning of the numbers rather than the quantity of the numbers.
Computer solvers can estimate the difficulty for a human to find the solution, based on the complexity of the solv ...
See also:Sudoku, Sudoku - Introduction, Sudoku - Rules and terminology, Sudoku - Solution methods, Sudoku - Scanning, Sudoku - Marking up, Sudoku - Analysis, Sudoku - Computer solutions, Sudoku - Difficulty ratings, Sudoku - Construction, Sudoku - Variants, Sudoku - Mathematics of Sudoku, Sudoku - History, Sudoku - Popularity in the media Read more here: » Sudoku: Encyclopedia II - Sudoku - Difficulty ratings |
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| | | | | OEIS: Encyclopedia II - Prime number theorem - Statement of the theorem Let π(x) be the prime counting function that gives the number of primes less than or equal to x, for any real number x. For example, π(10) = 4 because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states that the limit of the quotient of the two functions π(x) and x / ln(x) as x approaches infinity is 1. Using Landau notation this result can be written as
.
This does not mean that the limit of the difference< ...
See also:Prime number theorem, Prime number theorem - Statement of the theorem, Prime number theorem - The prime counting function in terms of the logarithmic integral, Prime number theorem - The issue of depth, Prime number theorem - The prime number theorem for arithmetic progressions, Prime number theorem - Bounds on the prime counting function, Prime number theorem - Approximations for the nth prime number, Prime number theorem - Gaps between primes, Prime number theorem - Table of πx x / ln x and Lix, Prime number theorem - Analogue for irreducible polynomials over a finite field Read more here: » Prime number theorem: Encyclopedia II - Prime number theorem - Statement of the theorem |
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| | | | | OEIS: Encyclopedia II - Sudoku - Construction It is possible to set starting grids with more than one solution and to set grids with no solution, but such are not considered proper Sudoku puzzles; like most other pure-logic puzzles, a unique solution is expected.
Building a Sudoku puzzle by hand can be performed efficiently by pre-determining the locations of the givens and assigning them values only as needed to make deductive progress. Such an undefined given can be assumed to not hold any particular value as long as it is given a different value before constructi ...
See also:Sudoku, Sudoku - Introduction, Sudoku - Rules and terminology, Sudoku - Solution methods, Sudoku - Scanning, Sudoku - Marking up, Sudoku - Analysis, Sudoku - Computer solutions, Sudoku - Difficulty ratings, Sudoku - Construction, Sudoku - Variants, Sudoku - Mathematics of Sudoku, Sudoku - History, Sudoku - Popularity in the media Read more here: » Sudoku: Encyclopedia II - Sudoku - Construction |
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| | | | | OEIS: Encyclopedia II - Sudoku - Introduction The name Sudoku is the Japanese abbreviation of a longer phrase, "suji wa dokushin ni kagiru," meaning "the digits must remain single"; it is a trademark of puzzle publisher Nikoli Co. Ltd in Japan. Other Japanese publishers refer to the puzzle as Nanpure (Number Place), the original U. S. title. In Japanese, the word is pronounced [sɯːdokɯ]; in English, it is usually spoken with an Anglicised pronunciation, [s ...
See also:Sudoku, Sudoku - Introduction, Sudoku - Rules and terminology, Sudoku - Solution methods, Sudoku - Scanning, Sudoku - Marking up, Sudoku - Analysis, Sudoku - Computer solutions, Sudoku - Difficulty ratings, Sudoku - Construction, Sudoku - Variants, Sudoku - Mathematics of Sudoku, Sudoku - History, Sudoku - Popularity in the media Read more here: » Sudoku: Encyclopedia II - Sudoku - Introduction |
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| | | | | OEIS: Encyclopedia II - Riemann zeta function - Definition The Riemann zeta function ζ(s) is defined for any complex number s with real part > 1 by the Dirichlet series:
In the region {s in C: Re(s) > 1}, this infinite series converges and defines a function analytic in this region. Bernhard Riemann realized that the zeta function can be extended by analytic continuation in a unique way to a meromorphic function ζ(s) defined for all complex numbers s with s ≠ 1. It is this fu ...
See also:Riemann zeta function, Riemann zeta function - Definition, Riemann zeta function - Values at the integers, Riemann zeta function - Relationship to prime numbers, Riemann zeta function - Proving the Euler product formula, Riemann zeta function - An easier proof for the layperson, Riemann zeta function - The importance of the zeros of ζs, Riemann zeta function - Basic properties, Riemann zeta function - The Riemann zeta function as a Mellin transform, Riemann zeta function - Series expansions, Riemann zeta function - Globally convergent series, Riemann zeta function - Universality, Riemann zeta function - Applications, Riemann zeta function - Generalizations, Riemann zeta function - Zeta functions in fiction Read more here: » Riemann zeta function: Encyclopedia II - Riemann zeta function - Definition |
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| | | | | | OEIS: Encyclopedia II - Cyclic number - Form of cyclic numbers From the relation to unit fractions, it can be shown that cyclic numbers are of the form
where b is the number base (10 for decimal), and p is a prime that does not divide b. (Primes p that give cyclic numbers are called full reptend primes or long primes).
For example, the case b = 10, p = 7 gives the cyclic number 142857.
Not all values of p will yield a cyclic number using this formula; for example p=13 gives 076923076923. These failed cases will a ...
See also:Cyclic number, Cyclic number - Special cases, Cyclic number - Relation to recurring decimals, Cyclic number - Form of cyclic numbers, Cyclic number - Construction of cyclic numbers, Cyclic number - Other numeric bases, Cyclic number - External link Read more here: » Cyclic number: Encyclopedia II - Cyclic number - Form of cyclic numbers |
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| | | | | OEIS: Encyclopedia II - Prime number theorem - Bounds on the prime counting function The prime number theorem is an asymptotic result. Hence, it cannot be used to bound π(x).
However, some bounds on π(x) are known, for instance
The first inequality holds for all x ≥ 17 and the second one for x > 1.
Another useful bound is
...
See also:Prime number theorem, Prime number theorem - Statement of the theorem, Prime number theorem - The prime counting function in terms of the logarithmic integral, Prime number theorem - The issue of depth, Prime number theorem - The prime number theorem for arithmetic progressions, Prime number theorem - Bounds on the prime counting function, Prime number theorem - Approximations for the nth prime number, Prime number theorem - Gaps between primes, Prime number theorem - Table of πx x / ln x and Lix, Prime number theorem - Analogue for irreducible polynomials over a finite field Read more here: » Prime number theorem: Encyclopedia II - Prime number theorem - Bounds on the prime counting function |
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| | | | | OEIS: Encyclopedia II - Prime number theorem - The prime number theorem for arithmetic progressions Let πn,a(x) denote the number of primes in the arithmetic progression a, a + n, a + 2n, a + 3n, … less than x. Dirichlet and Legendre conjectured, and Vallée Poussin proved, that, if a and n are coprime, then
where φ(·) is the Euler's totient function. In other words, the primes are distributed ...
See also:Prime number theorem, Prime number theorem - Statement of the theorem, Prime number theorem - The prime counting function in terms of the logarithmic integral, Prime number theorem - The issue of depth, Prime number theorem - The prime number theorem for arithmetic progressions, Prime number theorem - Bounds on the prime counting function, Prime number theorem - Approximations for the nth prime number, Prime number theorem - Gaps between primes, Prime number theorem - Table of πx x / ln x and Lix, Prime number theorem - Analogue for irreducible polynomials over a finite field Read more here: » Prime number theorem: Encyclopedia II - Prime number theorem - The prime number theorem for arithmetic progressions |
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| | | | | | | | OEIS: Encyclopedia II - Fibonacci number - Applications The Fibonacci numbers are important in the run-time analysis of Euclid's algorithm to determine the greatest common divisor of two integers.
Yuri Matiyasevich was able to show that the Fibonacci numbers can be defined by a Diophantine equation, which led to his original solution of Hilbert's tenth problem.
The Fibonacci numbers occur in a formula about the diagonals of Pascal's triangle (see binomial coefficient).
Every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci nu ...
See also:Fibonacci number, Fibonacci number - Origins, Fibonacci number - The bee ancestry code, Fibonacci number - Relation to the golden ratio, Fibonacci number - Matrix form, Fibonacci number - Computation, Fibonacci number - Applications, Fibonacci number - Fibonacci numbers in nature, Fibonacci number - Identities, Fibonacci number - Common factors, Fibonacci number - Power series, Fibonacci number - Reciprocal sum constant, Fibonacci number - Generalizations, Fibonacci number - Vector space, Fibonacci number - Similar integer sequences, Fibonacci number - Other generalizations, Fibonacci number - Fibonacci primes, Fibonacci number - Fibonacci strings, Fibonacci number - Fiction, Fibonacci number - Journals Read more here: » Fibonacci number: Encyclopedia II - Fibonacci number - Applications |
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| | | | | OEIS: Encyclopedia II - Prime number theorem - Gaps between primes The prime number theorem says that the "average" length of the gap between a prime p and the next prime is ln p. Of course, the actual length of the gap might be much more or less than this. For instance, the gap between a pair of twin primes contains only one number. On the other hand, the error term in the prime number theorem implies an upper bound on the length of a gap: for every ε > 0, there is a number q such that the gap is smaller than εp for all primes p larger than q. This result has been improved steadily. Baker et al. (2001) proved that gap is at most See also: Prime number theorem, Prime number theorem - Statement of the theorem, Prime number theorem - The prime counting function in terms of the logarithmic integral, Prime number theorem - The issue of depth, Prime number theorem - The prime number theorem for arithmetic progressions, Prime number theorem - Bounds on the prime counting function, Prime number theorem - Approximations for the nth prime number, Prime number theorem - Gaps between primes, Prime number theorem - Table of πx x / ln x and Lix, Prime number theorem - Analogue for irreducible polynomials over a finite field Read more here: » Prime number theorem: Encyclopedia II - Prime number theorem - Gaps between primes |
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| | | | | | OEIS: Encyclopedia II - Prime number theorem - Analogue for irreducible polynomials over a finite field There is an analogue of the prime number theorem that describes the "distribution" of irreducible polynomials over a finite field; the form it takes is strikingly similar to the case of the classical prime number theorem.
To state it precisely, let F = GF(q) be the finite field with q elements, for some fixed q, and let Nn be the number of monic irreducible polynomials over F whose degree is equal to n. That is, we are looking at polynomials with coefficients ch ...
See also:Prime number theorem, Prime number theorem - Statement of the theorem, Prime number theorem - The prime counting function in terms of the logarithmic integral, Prime number theorem - The issue of depth, Prime number theorem - The prime number theorem for arithmetic progressions, Prime number theorem - Bounds on the prime counting function, Prime number theorem - Approximations for the nth prime number, Prime number theorem - Gaps between primes, Prime number theorem - Table of πx x / ln x and Lix, Prime number theorem - Analogue for irreducible polynomials over a finite field Read more here: » Prime number theorem: Encyclopedia II - Prime number theorem - Analogue for irreducible polynomials over a finite field |
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| | | | | | | OEIS: Encyclopedia II - Latin square - Orthogonal array representation If each entry of an n × n Latin square is written as a triple (r,c,s), where r is the row, c is the column, and s is the symbol, we obtain a set of n2 triples called the orthogonal array representation of the square. For example, the orthogonal array representation of the first Latin square displayed above is
{ (1,1,1),(1,2,2),( ...
See also:Latin square, Latin square - Orthogonal array representation, Latin square - Equivalence classes of Latin squares, Latin square - The number of Latin squares, Latin square - Examples, Latin square - Latin squares and mathematical puzzles Read more here: » Latin square: Encyclopedia II - Latin square - Orthogonal array representation |
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| | | | | OEIS: Encyclopedia II - Pi - π culture March 14 (3/14 in US date format) marks Pi Day which is celebrated by many lovers of π.
On July 22, Pi Approximation Day is celebrated (22/7 - in European date format - is a popular approximation of π).
355/113 (~3.1415929) is sometimes jokingly referred to as "not π, but an incredible simulation!"
Singer Kate Bush's recently released album "Aerial" contains a song titled "π," in which she sings π to over one hundred decimal places. Fans have discovered that she got some of them wrong[1], however, and actually misses twenty-two numbers. Fans are ca ...
See also:Pi, Pi - Properties, Pi - Formulae involving π, Pi - Geometry, Pi - Analysis, Pi - Continued fractions, Pi - Number theory, Pi - Dynamical systems and ergodic theory, Pi - Physics, Pi - Probability and statistics, Pi - History of π, Pi - Numerical approximations of π, Pi - Miscellaneous formulae, Pi - Less accurate approximations, Pi - Open questions, Pi - The nature of π, Pi - Fictional references, Pi - π culture, Pi - Memorizing Pi Read more here: » Pi: Encyclopedia II - Pi - π culture |
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