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Finch (2001, 2003) defines a -rough (or -jagged) number to be positive integer all of whose prime factors are greater than or equal to .Greene and Knuth define "unusual numbers" as numbers whose greatest prime factor is greater than or equal to , and these number are dubbed "-rough" or "-jagged" by Finch (2001, 2003). The first few unusual numbers are 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, ... (OEIS A063538), which turn out to not be so unusual after all (Greene and Knuth 1990, Finch 2001). The first few "usual" numbers are then 8, 12, 16, 18, 24, 27, 30, ... (OEIS A063539).The probability that the greatest prime factor of a random integer is greater than is (Schroeppel 1972).

An abundant number, sometimes also called an excessive number, is a positive integer for which(1)where is the divisor function and is the restricted divisor function. The quantity is sometimes called the abundance.A number which is abundant but for which all its proper divisors are deficient is called a primitive abundant number (Guy 1994, p. 46).The first few abundant numbers are 12, 18, 20, 24, 30, 36, ... (OEIS A005101).Every positive integer with is abundant. Any multiple of a perfect number or an abundant number is also abundant. Prime numbers are not abundant. Every number greater than 20161 can be expressed as a sum of two abundant numbers.There are only 21 abundant numbers less than 100, and they are all even.The first odd abundant number is(2)That 945 is abundant can be seen by computing(3)Define the density function(4)(correcting the expression in Finch 2003, p. 126) for a positive real number where gives the cardinal..

A product involving an infinite number of terms. Such products can converge. In fact, for positive , the product converges to a nonzero number iff converges.Infinite products can be used to define the cosine(1)gamma function(2)sine, and sinc function.They also appear in polygon circumscribing,(3)An interesting infinite product formula due to Euler which relates and the th prime is(4)(5)(Blatner 1997). Knar's formula gives a functional equation for the gamma function in terms of the infinite product(6)A regularized product identity is given by(7)(Muñoz Garcia and Pérez-Marco 2003, 2008).Mellin's formula states(8)where is the digamma function and is the gamma function.The following class of products(9)(10)(11)(12)(13)(Borwein et al. 2004, pp. 4-6), where is the gamma function, the first of which is given in Borwein and Corless (1999), can be done analytically. In particular, for ,(14)where (Borwein et..

Sphere tetrahedron picking is the selection of quadruples of of points corresponding to vertices of a tetrahedron with vertices on the surface of a sphere. random tetrahedra can be picked on a unit sphere in the Wolfram Language using the function RandomPoint[Sphere[], n, 4].Pick four points on a sphere. What is the probability that the tetrahedron having these points as polyhedron vertices contains the center of the sphere? In the one-dimensional case, the probability that a second point is on the opposite side of 1/2 is 1/2. In the two-dimensional case, pick two points. In order for the third to form a triangle containing the center, it must lie in the quadrant bisected by a line segment passing through the center of the circle and the bisector of the two points. This happens for one quadrant, so the probability is 1/4. Similarly, for a sphere the probability is one octant, or 1/8.Pick four points at random on the surface of a unit sphereusing(1)(2)(3)with..

Ball triangle picking is the selection of triples of points (corresponding to vertices of a general triangle) randomly placed inside a ball. random triangles can be picked in a unit ball in the Wolfram Language using the function RandomPoint[Ball[], n, 3].The distribution of areas of a triangle with vertices picked at random in a unit ball is illustrated above. The mean triangle area is(1)(Buchta and Müller 1984, Finch 2010). random triangles can be picked in a unit ball in the Wolfram Language using the function RandomPoint[Ball[], n, 3].The determination of the probability for obtaining an acute triangle by picking three points at random in the unit disk was generalized by Hall (1982) to the -dimensional ball. Buchta (1986) subsequently gave closed form evaluations for Hall's integrals. Let be the probability that three points chosen independently and uniformly from the -ball form an acute triangle, then (2)(3)These can be combined..

Let be the number of (0,1)-matrices with no adjacent 1s (in either columns or rows). For , 2, ..., is given by 2, 7, 63, 1234, ... (OEIS A006506).The hard square entropy constant is defined by(OEIS A085850). It is not known if this constanthas an exact representation.The quantity arises in statistical physics (Baxter et al. 1980, Pearce and Seaton 1988), and is known as the entropy per site of hard squares. A related constant known as the hard hexagon entropy constant can also be defined.

In database structures, two quantities are generally of interest: the average number of comparisons required to 1. Find an existing random record, and 2. Insert a new random record into a data structure. Some constants which arise in the theory of digital tree searching are(1)(2)(3)(4)(5)(6)(OEIS A065442 and A065443), where is a q-polygamma function. Erdős (1948) proved that is irrational, and is sometimes known as the Erdős-Borwein constant.The expected number of comparisons for a successful search is(7)(8)and for an unsuccessful search is(9)(10)(OEIS A086309 and A086310). Here is the Euler-Mascheroni constant, , , and are small-amplitude periodic functions, and lg is the base 2 logarithm.The variance for searching is(11)(12)(OEIS A086311) and for inserting is(13)(14)(OEIS A086312).The expected number of pairs of twin vacancies in a digital search tree is(15)where(16)(17)(OEIS A086313), which can also be written(18)(Flajolet..

Consider an (0, 1)-matrix such as(1)for . Call two elements adjacent if they lie in positions and , and , or and for some . Call the number of such arrays with no pairs of adjacent 1s. Equivalently, is the number of configurations of nonattacking kings on an chessboard with regular hexagonal cells.The first few values of for , 2, ... are 2, 6, 43, 557, 14432, ... (OEIS A066863).The hard square hexagon constant is then given by(2)(3)(OEIS A085851).Amazingly, is algebraic and is given by(4)where(5)(6)(7)(8)(9)(10)(11)(Baxter 1980, Joyce 1988ab).The variable can be expressed in terms of the tribonacci constant(12)where is a polynomial root, as(13)(14)(15)(T. Piezas III, pers. comm., Feb. 11, 2006).Explicitly, is the unique positive root(16)where denotes the th root of the polynomial in the ordering of the Wolfram Language...

Let be the probability that a random walk on a -D lattice returns to the origin. In 1921, Pólya proved that(1)but(2)for . Watson (1939), McCrea and Whipple (1940), Domb (1954), and Glasser and Zucker (1977) showed that(3)(OEIS A086230), where(4)(5)(6)(7)(8)(9)(OEIS A086231; Borwein and Bailey 2003, Ch. 2, Ex. 20) is the third of Watson's triple integrals modulo a multiplicative constant, is a complete elliptic integral of the first kind, is a Jacobi theta function, and is the gamma function.Closed forms for are not known, but Montroll (1956) showed that for ,(10)where(11)(12)and is a modified Bessel function of the first kind.Numerical values of from Montroll (1956) and Flajolet (Finch 2003) are given in the following table.OEIS3A0862300.3405374A0862320.1932065A0862330.1351786A0862340.1047157A0862350.08584498A0862360.0729126..

The average distance between two points chosen at random inside a unit cube (the case of hypercube line picking), sometimes known as the Robbins constant, is(1)(2)(3)(OEIS A073012; Robbins 1978, Le Lionnais 1983).The probability function as a function of line length, illustrated above, was found in (nearly) closed form by Mathai et al. (1999). After simplifying, correcting typos, and completing the integrals, gives the closed form(4)The first even raw moments for , 2, ... are 1, 1/2, 11/30, 211/630, 187/525, 3524083/6306300, ... (OEIS A160693 and A160694).Pick points on a cube, and space them as far apart as possible. The best value known for the minimum straight line distance between any two points is given in the following table. 51.118033988749861.0606601482100718190.86602540378463100.74999998333331110.70961617562351120.70710678118660130.70710678118660140.70710678118660150.625..

Let denote the partition lattice of the set . The maximum element of is(1)and the minimum element is(2)Let denote the number of chains of any length in containing both and . Then satisfies the recurrence relation(3)where and is a Stirling number of the second kind. The first few values of for , 2, ... are then 1, 1, 4, 32, 436, 9012, 262760, ... (OEIS A005121).Lengyel (1984) proved that the quotient(4)is bounded between two constants as , and Flajolet and Salvy (1990) improved the result of Babai and Lengyel (1992) to show that(5)(OEIS A086053).

Jackson's theorem is a statement about the error of the best uniform approximation to a real function on by real polynomials of degree at most . Let be of bounded variation in and let and denote the least upper bound of and the total variation of in , respectively. Given the function(1)then the coefficients(2)of its Fourier-Legendre series, where is a Legendre polynomial, satisfy the inequalities(3)Moreover, the Fourier-Legendre series of converges uniformly and absolutely to in .Bernstein (1913) strengthened Jackson's theorem to(4)A specific application of Jackson's theorem shows that if(5)then(6)

Given the closed interval with , let one-dimensional "cars" of unit length be parked randomly on the interval. The mean number of cars which can fit (without overlapping!) satisfies(1)The mean density of the cars for large is(2)(3)(4)(OEIS A050996). While the inner integral canbe done analytically,(5)(6)where is the Euler-Mascheroni constant and is the incomplete gamma function, it is not known how to do the outer one(7)(8)(9)where is the exponential integral. The slowly converging series expansion for the integrand is given by(10)(OEIS A050994 and A050995).In addition,(11)for all (Rényi 1958), which was strengthened by Dvoretzky and Robbins (1964) to(12)Dvoretzky and Robbins (1964) also proved that(13)Let be the variance of the number of cars, then Dvoretzky and Robbins (1964) and Mannion (1964) showed that(14)(15)(16)(OEIS A086245), where(17)(18)and the numerical value is due to Blaisdell and Solomon..

Plouffe's constants are numbers arising in summations of series related to where is a trigonometric function. Define the Iverson bracket function(1)Now define through(2)(3)then(4)(5)(6)(OEIS A086201).For(7)(8)the sum is (amazingly) given by(9)(10)(11)(OEIS A086202), where denotes the XOR of binary digits (Chowdhury 2001a; Finch 2003, p. 432). A related sum is given by(12)(13)(14)(OEIS A111953), where again denotes the XOR of binary digits (Chowdhury 2001b; Finch 2005, p. 20).Letting(15)(16)then(17)(18)(OEIS A049541).Plouffe asked if the above processes could be "inverted." He considered(19)(20)giving(21)(22)and(23)(24)giving(25)(26)and(27)(28)giving(29)(30)(31)(OEIS A086203), where the identity was conjecturedby Plouffe and proved by Borwein and Girgensohn (1995). is sometimes known as Plouffe's constant (Plouffe 1997), although this angle had arisen in the geometry of the..

Let be an arbitrary trigonometric polynomial(1)with real coefficients, let be a function that is integrable over the interval , and let the th derivative of be bounded in . Then there exists a polynomial for which(2)for all , where is the smallest constant possible, known as the th Favard constant. can be given explicitly by the sum(3)which can be written in terms of the Lerch transcendentas(4)These can be expressed by(5)where is the Dirichlet lambda function and is the Dirichlet beta function. Explicitly,(6)(7)(8)(9)(10)(11)(OEIS A050970 and A050971).

The constantswhere and are th and th order polynomials, and is the set of all rational functions with real coefficients.

Consider the sum(1)where the s are nonnegative and the denominators are positive. Shapiro (1954) asked if(2)for all . It turns out (Mitrinovic et al. 1993) that this inequality is true for all even and odd .Define(3)and let(4)Then Rankin (1958) proved that(5) can be computed by letting be the function convex hull of the functions(6)(7)Then(8)(OEIS A086277; Drinfeljd 1971).A modified sum was considered by Elbert (1973):(9)Consider(10)where(11)and let be the convex hull of(12)(13)Then(14)(OEIS A086278).

Define with positive as(1)Then(2)as increases, where the Shallit constant is(3)(OEIS A086276; Shallit 1995). In their solution,Grosjean and De Meyer (quoted in Shallit 1995) reduced the complexity of the problem.

Let be the set of complex analytic functions defined on an open region containing the set closure of the unit disk satisfying and . For each in , let be the supremum of all numbers such that there is a subregion in on which is one-to-one and such that contains a disk of radius . In 1925, Bloch (Conway 1989) showed that .Define Bloch's constant by(1)Ahlfors and Grunsky (1937) derived(2)Bonk (1990) proved that , which was subsequently improved to (Chen and Gauthier 1996; Xiong 1998; Finch 2003, p. 456).Ahlfors and Grunsky (1937) also conjectured that the upper limit is actually the value of ,(3)(4)(5)(OEIS A085508; Le Lionnais 1983).

Let be the error of the best uniform approximation to a real function on the interval by real polynomials of degree at most . If(1)then Bernstein showed that(2)He conjectured that the lower limit () was . However, this was disproven by Varga and Carpenter (1987) and Varga (1990), who computed(3)For rational approximations for and of degree and , D. J. Newman (1964) proved(4)for . Gonchar (1967) and Bulanov (1975) improved the lower bound to(5)Vjacheslavo (1975) proved the existence of positive constants and such that(6)(Petrushev 1987, pp. 105-106). Varga et al. (1993) conjectured and Stahl(1993) proved that(7)

Given a positive integer , let its prime factorization be written(1)Define the functions and by , , and(2)(3)The first few terms of are 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, ... (OEIS A051904), while the first few terms of are 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, ... (OEIS A051903).Then the average value of tends to(4)Here, the running average values are given by 1/2, 2/3, 3/4, 1, 1, 1, 1, 11/9, 13/10, 14/11, 5/4, 16/13, ... (OEIS A086195 and A086196).In addition, the ratio(5)where is the Riemann zeta function (Niven 1969).Niven (1969) also proved that(6)where Niven's constant is given by(7)(OEIS A033150). Here, the running average values are given by 1/2, 2/3, 3/4, 1, 1, 1, 1, 11/9, 13/10, 14/11, 5/4, 17/13, ... (OEIS A086197 and A086198).The continued fraction of Niven's constant is 1, 1, 2, 2, 1, 1, 4, 1, 1, 3, 4, 4, 8, 4, 1, ... (OEIS A033151). The positions at which the digits 1, 2, ... first occur in the continued fraction are 1, 3, 10, 7, 47, 41, 34,..

Mills' theorem states that there exists a real constant such that is prime for all positive integers (Mills 1947). While for each value of , there are uncountably many possible values of such that is prime for all positive integers (Caldwell and Cheng 2005), it is possible to define Mills' constant as the least such thatis prime for all positive integers , giving a value of(OEIS A051021). is therefore given by the next prime after , and the values of are known as Mills' primes (Caldwell and Cheng 2005).Caldwell and Cheng (2005) computed more than 6850 digits of assuming the truth of the Riemann hypothesis. Proof of primality of the 13 Mills prime in Jul. 2013 means that approximately digits are now known.It is not known if is irrational.

Let and be nonzero integers such that (except when ). Also let be the set of primes for which for some nonnegative integer . Then assuming the generalized Riemann hypothesis, Stephens (1976) showed that the density of relative to the primes is a rational multiple of(OEIS A065478), where is the th prime (Finch 2003).

(1)(2)(OEIS A093827), where is the totient function and is the divisor function.

The Mertens constant , also known as the Hadamard-de la Vallee-Poussin constant, prime reciprocal constant (Bach and Shallit 1996, p. 234), or Kronecker's constant (Schroeder 1997), is a constant related to the twin primes constant and that appears in Mertens' second theorem,(1)where the sum is over primes and is a Landau symbol. This sum is the analog of(2)where is the Euler-Mascheroni constant (Gourdon and Sebah).The constant is given by the infinite sum(3)where is the Euler-Mascheroni constant and is the th prime (Rosser and Schoenfeld 1962; Hardy and Wright 1979; Le Lionnais 1983; Ellison and Ellison 1985), or by the limit(4)According to Lindqvist and Peetre (1997), this was shown independently by Meisselin 1866 and Mertens (1874). Formula (3) is equivalent to(5)which follows from (4) using the Mercator series for with . is also given by the rapidly converging series(6)where is the Riemann zeta function, and is the Möbius..

The constant(OEIS A014715) giving the asymptotic rate of growth of the number of digits in the th term of the look and say sequence, given by the unique positive real root of the polynomialillustrated in the figure above. Note that the polynomialgiven in Conway (1987, p. 188) contains a misprint.The continued fraction for is 1, 3, 3, 2, 1, 2, 1, 5, 8, 4, 14, 3, 1, ... (OEIS A014967).

A "square" word consists of two identical adjacent subwords (for example, acbacb). A squarefree word contains no square words as subwords (for example, abcacbabcb). The only squarefree binary words are , , ab, ba, aba, and bab (since aa, bb, aaa, aab, abb, baa, bba, and bbb contain square identical adjacent subwords a, b, a, a, b, a, b, and b, respectively).However, there are arbitrarily long ternary squarefree words. The number of ternary squarefree words of length , 2, ... are 1, 3, 6, 12, 18, 30, 42, 60, ... (OEIS A006156), and is bounded by(Brandenburg 1983). In addition,(Brinkhuis 1983, Noonan and Zeilberger 1999).The number of squarefree quaternary words of length , 2, ... are 4, 12, 36, 96, 264, 696, ... (OEIS A051041).

A word is said to be overlapfree if it has no subwords of the form xyxyx. A squarefree word is overlapfree, and an overlapfree word is cubefree.The numbers of binary overlapfree words of length , 2, ... are 2, 4, 6, 10, 14, 20, ... (OEIS A007777). satisfies(1)for some constants and (Restivo and Selemi 1985, Kobayashi 1988). In addition, while(2)does not exist,(3)where(4)(5)(Cassaigne 1993).The Thue-Morse sequence is overlapfree (Alloucheand Shallit 2003, p. 15).

A perfect power is a number of the form , where is a positive integer and . If the prime factorization of is , then is a perfect power iff .Including duplications (i.e., taking all numbers up to some cutoff and taking all their powers) and taking , the first few are 4, 8, 9, 16, 16, 25, 27, 32, 36, 49, 64, 64, 64, ... (OEIS A072103). Here, 16 is duplicated since(1)As shown by Goldbach, the sum of reciprocals of perfect powers (excluding 1) withduplications converges,(2)The first few numbers that are perfect powers in more than one way are 16, 64, 81,256, 512, 625, 729, 1024, 1296, 2401, 4096, ... (OEIS A117453).The first few perfect powers without duplications are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, ... (OEIS A001597). Even more amazingly, the sum of the reciprocals of these numbers (excluding 1) is given by(3)(OEIS A072102), where is the Möbius function and is the Riemann zeta function.The numbers of perfect powers without duplications..

The Heilbronn triangle problem is to place points in a disk (square, equilateral triangle, etc.) of unit area so as to maximize the area of the smallest of the triangles determined by the points. For points, there is only a single triangle, so Heilbronn's problem degenerates into finding the largest triangle that can be constructed from points in a square. For , there are four possible triangles for each configuration, so the problem is to find the configuration of points for which the smallest of these four triangles is the maximum possible.For a unit square, the first few maxima of minimaltriangle areas are(1)(2)(3)(4)(5)(6)(7)(8)For larger values of , proofs of optimality are open, but the best known results are(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)with the configurations leading to maximum minimal triangles illustrated above (Friedman 2006; Comellas and Yebra 2002; D. Cantrell..

Apéry's constant is defined by(1)(OEIS A002117) where is the Riemann zeta function. Apéry (1979) proved that is irrational, although it is not known if it is transcendental. Sorokin (1994) and Nesterenko (1996) subsequently constructed independent proofs for the irrationality of (Hata 2000). arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio, computed using quantum electrodynamics.The following table summarizes progress in computing upper bounds on the irrationality measure for . Here, the exact values for is given by(2)(3)(Hata 2000).upper boundreference15.513891Rhin and Viola (2001)28.830284Hata (1990)312.74359Dvornicich and Viola (1987)413.41782Apéry (1979), Sorokin (1994), Nesterenko (1996), Prévost (1996)Beukers (1979) reproduced Apéry's rational approximation to using the triple..

A generalization of the equation whose solution is desired in Fermat'slast theoremtofor , , , and positive constants, with trivial solutions having , , or being excluded. is trivial to solve by taking and . is more difficult, but can be solved by noting that solutions exist for values of which can be written as a sum of two squares, the first few of which are 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, ... (OEIS A001481).

Let(1)be the simple continued fraction of a "generic" real number , where the numbers are the partial quotients. Khinchin (1934) considered the limit of the geometric mean(2)as . Amazingly, except for a set of measure 0, this limit is a constant independent of given by(3)(OEIS A002210), as proved in Kac (1959).The constant is known as Khinchin's constant, and is commonly also spelled "Khintchine'sconstant" (Shanks and Wrench 1959, Bailey et al. 1997).It is implemented as Khinchin, where its value is cached to 1100-digit precision. However, the numerical value of is notoriously difficult to calculate to high precision, so computation of more digits get increasingly slower.It is not known if is irrational, let alone transcendental.While it is known that almost all numbers have limits that approach , this fact has not been proven for any explicit real number , e.g., a real number cast in terms of fundamental constants..

A number of closed-form constants can be obtained for generalized continued fractions having particularly simple partial numerators and denominators.The Ramanujan continued fractions provide a fascinating class of continued fraction constants. The Trott constants are unexpected constants whose partial numerators and denominators correspond to their decimal digits (though to achieve this, it is necessary to allow some partial numerators to equal 0).The first in a series of other famous continued fraction constants is the infiniteregular continued fraction(1)(2)The first few convergents of the constant are 0, 1, 2/3, 7/10, 30/43, 157/225, 972/1393, 6961/9976, ... (OEIS A001053 and A001040).Both numerator and denominator satisfy the recurrence relation(3)where has the initial conditions , and has the initial conditions , . These can be solved exactly to yield(4)(5)(6)(7)where is a modified Bessel function of the first kind..

A weakly binary tree is a planted tree in which all nonroot graph vertices are adjacent to at most three graph vertices.Let(1)be the generating function for the number of weakly binary trees on nodes, where(2)(3)(4)(5)This gives the sequence 1, 1, 1, 2, 3, 6, 11, 23, 46, 98, 207, ... (OEIS A001190),sometimes also known as the Wedderburn-Etherington numbers.Otter (Otter 1948, Harary and Palmer 1973, Knuth 1997) showed that(6)where(7)(OEIS A086317; Knuth 1997, p. 583; Finch 2003, p. 297) is the unique positive root of(8)and(9)(OEIS A086318; Knuth 1997, p. 583). is also given by the rapidly converging limit(10)where is given by(11)(12)the first few terms of which are 6, 38, 1446, 2090918, 4371938082726, ... (OEIS A072191), giving(13)

The Lehmer cotangent expansion for whichthe convergence is slowest occurs when the inequality in the recurrence equation(1)for(2)is replaced by equality, giving and(3)for .This recurrences gives values of corresponding to 0, 1, 3, 13, 183, 33673, ... (OEIS A002065), and defines the constant known as Lehmer's constant as(4)(5)(6)(OEIS A030125). is not an algebraic number of degree less than 4, but Lehmer's approach cannot show whether is transcendental.

Expanding the Riemann zeta function about gives(1)(Havil 2003, p. 118), where the constants(2)are known as Stieltjes constants.Another sum that can be used to define the constants is(3)These constants are returned by the WolframLanguage function StieltjesGamma[n].A generalization takes as the coefficient of is the Laurent series of the Hurwitz zeta function about . These generalized Stieltjes constants are implemented in the Wolfram Language as StieltjesGamma[n, a].The case gives the usual Euler-Mascheroni constant(4)A limit formula for is given by(5)where is the imaginary part and is the Riemann zeta function.An alternative definition is given by absorbing the coefficient of into the constant,(6)(e.g., Hardy 1912, Kluyver 1927).The Stieltjes constants are also given by(7)Plots of the values of the Stieltjes constants as a function of are illustrated above (Kreminski). The first few numerical values are given in the..

Let be defined as the power series whose th term has a coefficient equal to the th prime ,(1)(2)The function has a zero at (OEIS A088751). Now let be defined by(3)(4)(5)(OEIS A030018).Then N. Backhouse conjectured that(6)(7)(OEIS A072508). This limit was subsequently shown to exist by P. Flajolet. Note that , which follows from the radius of convergence of the reciprocal power series.The continued fraction of Backhouse's constant is [1, 2, 5, 5, 4, 1, 1, 18, 1, 1, 1, 1, 1, 2, ...] (OEIS A074269), which is also the same as the continued fraction of except for a leading 0 in the latter.

There are two sets of constants that are commonly known as Lebesgue constants. The first is related to approximation of function via Fourier series, which the other arises in the computation of Lagrange interpolating polynomials.Assume a function is integrable over the interval and is the th partial sum of the Fourier series of , so that(1)(2)and(3)If(4)for all , then(5)and is the smallest possible constant for which this holds for all continuous . The first few values of are(6)(7)(8)(9)(10)(11)(12)(13)Some sum formulas for include(14)(15)(Zygmund 1959) and integral formulas include(16)(17)(Hardy 1942). For large ,(18)This result can be generalized for an -differentiable function satisfying(19)for all . In this case,(20)where(21)(Kolmogorov 1935, Zygmund 1959).Watson (1930) showed that(22)where(23)(24)(25)(OEIS A086052), where is the gamma function, is the Dirichlet lambda function, and is the Euler-Mascheroni constant.Define..

A sequence of positive integers(1)is a nonaveraging sequence if it contains no three terms which are in an arithmeticprogression, i.e., terms such that(2)for distinct , , . The empty set and sets of length one are therefore trivially nonaveraging.Consider all possible subsets on the integers . There is one nonaveraging sequence on (), two on ( and ), four on , and so on. For example, 13 of the 16 subjects of are nonaveraging, with , , and excluded. The numbers of nonaveraging subsets on , , ... are 1, 2, 4, 7, 13, 23, 40, ... (OEIS A051013).Wróblewski (1984) showed that for infinite nonaveraging sequences,(3)

Let be a real number, and let be the set of positive real numbers for which(1)has (at most) finitely many solutions for and integers. Then the irrationality measure, sometimes called the Liouville-Roth constant or irrationality exponent, is defined as the threshold at which Liouville's approximation theorem kicks in and is no longer approximable by rational numbers,(2)where is the infimum. If the set is empty, then is defined to be , and is called a Liouville number. There are three possible regimes for nonempty :(3)where the transitional case can correspond to being either algebraic of degree or being transcendental. Showing that for an algebraic number is a difficult result for which Roth was awarded the Fields medal.The definition of irrationality measure is equivalent to the statement that if has irrationality measure , then is the smallest number such that the inequality(4)holds for any and all integers and with sufficiently large.The..

Let be a positive nonsquare integer. Then Artin conjectured that the set of all primes for which is a primitive root is infinite. Under the assumption of the generalized Riemann hypothesis, Artin's conjecture was solved by Hooley (1967; Finch 2003, p. 105).Let be not an th power for any such the squarefree part of satisfies (mod 4). Let be the set of all primes for which such an is a primitive root. Then Artin also conjectured that the density of relative to the primes is given independently of the choice of by , where(1)(OEIS A005596), and is the th prime.The significance of Artin's constant is more easily seen by describing it as the fraction of primes for which has a maximal period repeating decimal, i.e., is a full reptend prime (Conway and Guy 1996) corresponding to a cyclic number. is connected with the prime zeta function by(2)where is a Lucas number (Ribenboim 1998, Gourdon and Sebah). Wrench (1961) gave 45 digits of , and Gourdon and Sebah..

What is the sofa of greatest area which can be moved around a right-angled hallway of unit width? Hammersley (Croft et al. 1994) showed that(1)(OEIS A086118). Gerver (1992) found a sofa with larger area and provided arguments indicating that it is either optimal or close to it. The boundary of Gerver's sofa is a complicated shape composed of 18 arcs. Its area can be given by defining the constants , , , and by solving(2)(3)(4)(5)This gives(6)(7)(8)(9)Now define(10)where(11)(12)(13)Finally, define the functions(14)(15)(16)The area of the optimal sofa is then given by(17)(18)(Finch 2003).

The Hermite constant is defined for dimension as the value(1)(Le Lionnais 1983). In other words, they are given by(2)where is the maximum lattice packing density for hypersphere packing and is the content of the -hypersphere. The first few values of are 1, 4/3, 2, 4, 8, 64/3, 64, 256, ... (OEIS A007361 and A007362; Gruber and Lekkerkerker 1987, p. 518). Values for larger are not known.For sufficiently large ,(3)

Given a unit disk, find the smallest radius required for equal disks to completely cover the unit disk. The first few such values are(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)Here, values for , 8, 9, 10 are approximate values obtained using computer experimentation by Zahn (1962).For a symmetrical arrangement with (known as the five disks problem), , where is the golden ratio. However, rather surprisingly, the radius can be slightly reduced in the general disk covering problem where symmetry is not required; this configuration is illustrated above (Friedman). Neville (1915) showed that the value is equal to , where and are solutions to(11)(12)(13)(14)These solutions can be found exactly as(15)(16)where (17)(18)are the smallest positive roots of the given polynomials, with denoting the th root of the polynomial in the ordering of the Wolfram Language. This gives (OEIS A133077) exactly as(19)where the root is the smallest positive one of the..

Closed forms are known for the sums of reciprocals of even-indexed Fibonaccinumbers(1)(2)(3)(4)(5)(6)(7)(OEIS A153386; Knopp 1990, Ch. 8, Ex. 114; Paszkowski 1997; Horadam 1988; Finch 2003, p. 358; E. Weisstein, Jan. 1, 2009; Arndt 2012), where is the golden ratio, is a q-polygamma function, and is a Lambert series (Borwein and Borwein 1987, pp. 91 and 95) and odd-indexed Fibonacci numbers(8)(9)(10)(11)(12)(13)(OEIS A153387; Landau 1899; Borwein and Borwein 1997, p. 94; E. Weisstein, Jan. 1, 2009; Arndt 2012), where is a Jacobi elliptic function. Together, these give a closed form for the reciprocal Fibonacci constant of(14)(15)(16)(17)(18)(OEIS A079586; Horadam 1988; Griffin 1992; Zhao 1999; Finch 2003, p. 358). The question of the irrationality of was formally raised by Paul Erdős and this sum was proved to be irrational by André-Jeannin (1989).Borwein..

The case of the Weierstrass elliptic function with invariants and . The corresponding real half-period is given by(1)(2)(OEIS A064582), known as the omega2-constant, where is the gamma function. The other half-period is then given by(3)(4)(5)(OEIS A094961 and A094962).

The power tower of order is defined as(1)where is Knuth up-arrow notation (Knuth 1976), which in turn is defined by(2)together with(3)(4)Rucker (1995, p. 74) uses the notation(5)and refers to this operation as "tetration."A power tower can be implemented in the WolframLanguage as PowerTower[a_, k_Integer] := Nest[Power[a, #]&, 1, k]or PowerTower[a_, k_Integer] := Power @@ Table[a, {k}]The following table gives values of for , 2, ... for small .OEIS1A0000271, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...2A0003121, 4, 27, 256, 3125, 46656, ...3A0024881, 16, 7625597484987, ...41, 65536, ...The following table gives for , 2, ... for small .OEIS1A0000121, 1, 1, 1, 1, 1, ...2A0142212, 4, 16, 65536, , ...3A0142223, 27, 7625597484987, ...44, 256, , ...Consider and let be defined as(6)(Galidakis 2004). Then for , is entire with series expansion:(7)Similarly, for , is analytic for in the domain of the principal branch of , with series expansion:(8)For..

The Euclidean algorithm, also called Euclid's algorithm, is an algorithm for finding the greatest common divisor of two numbers and . The algorithm can also be defined for more general rings than just the integers . There are even principal rings which are not Euclidean but where the equivalent of the Euclidean algorithm can be defined. The algorithm for rational numbers was given in Book VII of Euclid's Elements. The algorithm for reals appeared in Book X, making it the earliest example of an integer relation algorithm (Ferguson et al. 1999).The Euclidean algorithm is an example of a P-problem whose time complexity is bounded by a quadratic function of the length of the input values (Bach and Shallit 1996).Let , then find a number which divides both and (so that and ), then also divides since(1)Similarly, find a number which divides and (so that and ), then divides since(2)Therefore, every common divisor of and is a common divisor of and , so the procedure..

Porter's constant is the constant appearing in formulasfor the efficiency of the Euclidean algorithm,(1)(2)(3)(OEIS A086237), where is the Euler-Mascheroni constant, is the Riemann zeta function, and is the Glaisher-Kinkelin constant (Knuth 1998, p. 357). The notation is generally used for this constant (Knuth 1998, p. 357, Finch 2003, pp. 156-157), though other authors use (Ustinov 2010) or (Dimitrov et al. 2000).The related constant originally considered by Porter (1975) and Knuth (1976) was denoted and , respectively, and defined by(4)(5)Knuth (1976) suggested be called the Lochs-Porter constant due to the work of Lochs (1961).

There are essentially three types of Fisher-Tippett extreme value distributions. The most common is the type I distribution, which are sometimes referred to as Gumbel types or just Gumbel distributions. These are distributions of an extreme order statistic for a distribution of elements .The Fisher-Tippett distribution corresponding to a maximum extreme value distribution (i.e., the distribution of the maximum ), sometimes known as the log-Weibull distribution, with location parameter and scale parameter is implemented in the Wolfram Language as ExtremeValueDistribution[alpha, beta].It has probability density functionand distribution function(1)(2)The moments can be computed directly by defining(3)(4)(5)Then the raw moments are(6)(7)(8)(9)(10)(11)where are Euler-Mascheroni integrals. Plugging in the Euler-Mascheroni integrals gives(12)(13)(14)(15)(16)where is the Euler-Mascheroni constant and is Apéry's..

An Abelian group is a group for which the elements commute (i.e., for all elements and ). Abelian groups therefore correspond to groups with symmetric multiplication tables.All cyclic groups are Abelian, but an Abelian group is not necessarily cyclic. All subgroups of an Abelian group are normal. In an Abelian group, each element is in a conjugacy class by itself, and the character table involves powers of a single element known as a group generator.In the Wolfram Language, the function AbelianGroup[n1, n2, ...] represents the direct product of the cyclic groups of degrees , , ....No general formula is known for giving the number of nonisomorphic finite groups of a given group order. However, the number of nonisomorphic Abelian finite groups of any given group order is given by writing as(1)where the are distinct prime factors, then(2)where is the partition function, which is implemented in the Wolfram Language as FiniteAbelianGroupCount[n]...

Let be a compact connected subset of -dimensional Euclidean space. Gross (1964) and Stadje (1981) proved that there is a unique real number such that for all , , ..., , there exists with(1)The magic constant of is defined by(2)where(3)These numbers are also called dispersion numbers and rendezvous values. For any , Gross (1964) and Stadje (1981) proved that(4)If is a subinterval of the line and is a circular disk in the plane, then(5)If is a circle, then(6)(OEIS A060294). An expression for the magic constant of an ellipse in terms of its semimajor and semiminor axes lengths is not known. Nikolas and Yost (1988) showed that for a Reuleaux triangle (7)Denote the maximum value of in -dimensional space by . Thenwhere is the gamma function (Nikolas and Yost 1988).An unrelated quantity characteristic of a given magicsquare is also known as a magic constant...

Let be the set of complex analytic functions defined on an open region containing the closure of the unit disk satisfying and . For each in , let be the supremum of all numbers such that contains a disk of radius . ThenThis constant is called the Landau constant, or the Bloch-Landau constant. Robinson (1938, unpublished) and Rademacher (1943) derived the bounds(OEIS A081760), where is the gamma function, and conjectured that the second inequality is actually an equality.

Define the sequence , , and(1)for . The first few values are(2)(3)(4)(5)Janssen and Tjaden (1987) showed that this sequence converges for exactly one value , where (OEIS A085835), confirming Grossman's conjecture. However, no analytic form is known for this constant, either as the root of a function or as a combination of other constants. The plot above shows the first few iterations of for to 30, with odd shown in red and even shown in blue, for ranging from 0 to 1. As can be seen, the solutions alternate by parity. For each fixed , the red values go to 0, while the blue values go to some positive number.Nyerges (2000) has generalized the recurrence to the functional equation(6)

A quadratic recurrence is a recurrence equation on a sequence of numbers expressing as a second-degree polynomial in with . For example,is a quadratic recurrence equation.A quadratic recurrence equation of the formin which no cross terms are present is known as a quadraticmap.

The de Bruijn constant, also called the Copson-de Bruijn constant, is the minimal constant(OEIS A113276) such that the inequalityalways holds.

Extend Hilbert's inequality by letting and(1)so that(2)Levin (1937) and Stečkin (1949) showed that(3)and(4)Mitrinovic et al. (1991) indicate that this constant is the best possible.

Let be the smallest tour length for points in a -D hypercube. Then there exists a smallest constant such that for all optimal tours in the hypercube,(1)and a constant such that for almost all optimal tours in the hypercube,(2)These constants satisfy the inequalities(3)(4)(5)(6)(7)(8)(9)(Fejes Tóth 1940, Verblunsky 1951, Few 1955, Beardwood et al. 1959),where(10) is the gamma function, is an expression involving Struve functions and Bessel functions of the second kind,(11)(OEIS A086306; Karloff 1989), and(12)(OEIS A086307; Goddyn 1990).In the limit ,(13)(14)(15)and(16)where(17)and is the best sphere packing density in -D space (Goddyn 1990, Moran 1984, Kabatyanskii and Levenshtein 1978). Steele and Snyder (1989) proved that the limit exists.Now consider the constant(18)so(19)Nonrigorous numerical estimates give (Johnson et al. 1996) and (Percus and Martin 1996).A certain self-avoiding space-filling function..

Let be a number field with real embeddings and imaginary embeddings and let . Then the multiplicative group of units of has the form(1)where is a primitive th root of unity, for the maximal such that there is a primitive th root of unity of . Whenever is quadratic, (unless , in which case , or , in which case ). Thus, is isomorphic to the group . The generators for are called the fundamental units of . Real quadratic number fields and imaginary cubic number fields have just one fundamental unit and imaginary quadratic number fields have no fundamental units. Observe that is the order of the torsion subgroup of and that the are determined up to a change of -basis and up to a multiplication by a root of unity.The fundamental unit of a number field is intimatelyconnected with the regulator.The fundamental units of a field generated by the algebraic number can be computed in the Wolfram Language using NumberFieldFundamentalUnits[a].In a real quadratic field,..

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