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Mathematical constants. (English) Zbl 1054.00001

Encyclopedia of Mathematics and Its Applications 94. Cambridge: Cambridge University Press (ISBN 0-521-81805-2/hbk). xx, 602 p. (2003).
There are topics that go across all fields of mathematics. Numerical constants are one of them and they are here, for the first time, the object of a systematic investigation. “The emphasis is not on the decimal expansions, but rather on the mathematical origins of the constants and their interrelationships. In short, the stories, not the table, tie the book together.” “My intended audience is advanced undergraduates and beyond (so I may assume readers have had calculus, matrix theory, differential equations, probability, some abstract algebra, and analysis)”.
The book is organized in 8 chapters, a table of constants, an author index, a subject index,
Chapter 1 is devoted to famous constants, some of them occurring in elementary mathematics: the Golden Mean, the natural logarithmic base, Pythagoras’ \(\sqrt{2}\), Archimedes’ \(\pi\), Euler-Mascheroni’s \(\gamma\), Apéry’s \(\zeta(3)\), Catalan’s \(G\), and the constants considered by Khintchine-Lévy, Feigenbaum-Coullet-Tresser, Madelung and Chaitin.
Chapter 2 refers to constants associated with number theory, involving names such as Hardy-Littlewood, Meissel-Mertens, Landau-Ramanujan, Artin, Hafner-Sarnak-McCurley, Niven, Euler, Pell-Stevenhagen, Alladi-Grinstead, Sierpinski, Linnik, Mills, Brun, Glaisher-Kinkelin,
Stolarsky-Harborth, Ulam, Gauss-Kuzmin-Wirsing, Porter-Hensley, Vallée, Erdős, Stieltjes, Liouville-Roth, Cameron, Pisot, Freiman, DeBruijn-Newman, Hall-Montgomery.
Chapter 3 is devoted to constants associated with analytic irregularities and involve names such as: Shapiro-Drinfeld, Carlson-Levin, Landau-Kolmogorov, Hilbert, Sobolev, Korn, Whitney-Mikhlin, Zolotarev-Schur, Kneser-Mahler, Grothendieck, DuBois Reymond, Steinitz, Young-Fejér-Jackson, Van der Corput, Turán.
Constants associated with approximation of functions are considered in Chapter 4 and involve names such as: Gibbs-Wilbraham, Lebesgue, Akhiezer-Krein-Favard, Bernstein, Fransén-Robinson, Berry-Esseen, Laplace, Chebyshev.
In Chapter 5, related to constants associated to enumerating discrete structures, involve names such as: Rényi, Golomb-Dickman, Kalmár, Otter, Lengyel, Pólya, Feller, Klarner, Lieb. Tutte.
Constants associated with functional iteration (Chapter 6) involve names such as: Gauss, Weierstrass, Euler-Gompertz, Kepler-Bouwkamp, Cahen, Lehmer, Plouffe, Grossman, Prouhet-Thue-Morse, Minkowski-Bower, Conway.
In complex analysis (Chapter 7): Bloch-Landau, Masser-Gramain, Whittaker-Goncharov, John, Hayman, Littlewood-Clunie-Pommerenke, Riesz-Kolmogorov, Grötzsch.
In geometry (Chapter 8): Moser, Sterner, Hermite, Calabi, DeVicci, Graham, Heilbronn, Kakeya-Besicovitch.
This book is a valuable source of information and of suggestions for further research. A natural continuation could be related to mathematical constants in applied fields, such as mathematical computer science, probability, mathematical physics etc. Important aspects of human relations point out the meaning and the significance of some natural numbers, such as 1, 2, 3, 4, 5, 7 (see G. Miller’s “magical number seven”) etc. Finch’s book enables to develop bald analogies, such as between the Conway’s constant and the golden ratio.

MSC:

00A20 Dictionaries and other general reference works
00A05 Mathematics in general
11Y60 Evaluation of number-theoretic constants
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Online Encyclopedia of Integer Sequences:

Number of n-node unrooted quartic trees; number of n-carbon alkanes C(n)H(2n+2) ignoring stereoisomers.
Decimal expansion of Pi (or digits of Pi).
Decimal expansion of Euler’s constant (or the Euler-Mascheroni constant), gamma.
Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2.
Number of irreducible polynomials of degree n over GF(5); dimensions of free Lie algebras.
Apéry’s number or Apéry’s constant zeta(3). Decimal expansion of zeta(3) = Sum_{m >= 1} 1/m^3.
Decimal expansion of square root of 2.
Decimal expansion of Khintchine’s constant.
a(n) = Product_{i=1..n} (2^i - 1). Also called 2-factorial numbers.
Decimal expansion of the twin prime constant C_2 = Product_{ p prime >= 3 } (1-1/(p-1)^2).
a(n) = (a(n-1) + 1)*a(n-2).
Decimal expansion of Catalan’s constant 1 - 1/9 + 1/25 - 1/49 + 1/81 - ...
Decimal expansion of Feigenbaum bifurcation velocity.
Decimal expansion of Feigenbaum reduction parameter.
Decimal expansion of log_2 e.
Number of B-trees of order 3 with n leaves.
Consider the Morse-Thue sequence (A010060) as defining a binary constant and convert it to decimal.
Odd cubes: a(n) = (2*n + 1)^3.
Decimal expansion of Pi*e/2.
Decimal expansion of sqrt(2*Pi*e).
Decimal expansion of sqrt(2*Pi).
Decimal expansion of 1/sqrt(80) = sqrt(5)/20.
Decimal expansion of zeta(2)*zeta(3)*zeta(4)*...
Decimal expansion of Niven’s constant.
Decimal expansion of Integral_{x=0..Pi} (sin(x)/x) dx.
Decimal expansion of (2/Pi)*Integral_{x=0..Pi} sin(x)/x dx.
Decimal expansion of Calabi’s constant.
a(0) = 1; thereafter a(n) = n*a(n-1)^2.
Decimal expansion of Komornik-Loreti constant.
Decimal expansion of area under the curve 1/Gamma(x) from zero to infinity.
Decimal expansion of Buffon’s constant 2/Pi.
Decimal expansion of the continued fraction constant (base 10).
Real half-period for the Weierstrass elliptic function with invariants g2=0, g3=1.
Decimal expansion of Product_{p prime} (1 - p/(p^3-1)).
Decimal expansion of Soldner’s constant.
Decimal expansion of sum of reciprocal perfect powers (excluding 1).
Decimal expansion of Pi^2/12.
Decimal expansion of Robbins constant.
Decimal expansion of the mean number of iterations in comparing two numbers via their continued fractions.
Decimal expansion of Glaisher-Kinkelin constant A.
Decimal expansion of first solution of equation cos(x) cosh(x) = 1.
Stolarsky-Harborth constant; lim inf_{n->oo} F(n)/n^theta, where F(n) is the number of odd binomial coefficients in the first n rows and theta=log(3)/log(2).
Decimal expansion of lim_{n -> infinity} A001699(n)^(1/2^n).
Decimal expansion of Mertens’s constant, which is the limit of (Sum_{i=1..k} 1/prime(i)) - log(log(prime(k))) as k goes to infinity, where prime(i) is the i-th prime number.
Decimal expansion of sum of alternating series of reciprocals of primes.
Decimal expansion of constant C such that Sum_{k>=1} 1/C^p(k) = 1 where p(k) is the k-th prime.
Decimal expansion of Sum_{k>=1} 1/F(k) where F(k) is the k-th Fibonacci number A000045(k).
Decimal expansion of exp(-gamma).
Decimal expansion of average deviation of the total number of prime factors.
Decimal expansion of constant B3 (or B_3) related to the Mertens constant.
Decimal expansion of Product_{j>=1, j!=2} zeta(j/2) (negated).
Decimal expansion of Product_{j>=1, j!=3} zeta(j/3).
Decimal expansion of linear asymptotic constant B in Sum_{k=1..n} 1/A000688(k) =  B*n + ...
Decimal expansion of Golomb-Dickman constant.
Decimal expansion of Alladi-Grinstead constant exp(c-1), where c is given in A085361.
Decimal expansion of the number c = Sum_{n>=1} (zeta(n+1)-1)/n.
Decimal expansion of the Kepler-Bouwkamp or polygon-inscribing constant.
Decimal expansion of conjectured value for the Bloch constant.
Decimal expansion of the prime zeta function at 2: Sum_{p prime} 1/p^2.
Decimal expansion of Sum{p prime>=2} log(p)/(p^2-p+1).
Decimal expansion of the probability that two m X m and n X n matrices (m,n large) have relatively prime determinants.
Decimal expansion of Pi/2 + 2/Pi.
Decimal expansion of arctan(1/2)/Pi.
Decimal expansion of probability that a random walk on a 3-D lattice returns to the origin.
Decimal expansion of Porter’s Constant.
Decimal expansion of Sum_{k>=2} c(k)/prime(k), where c(k) = -1 if p == 1 (mod 4) and c(k) = +1 if p == 3 (mod 4).
Decimal expansion of Sum_{k>=2} (p mod 4 - 2)/p^2 where p=prime(k).
Decimal expansion of value to which Sum_{k>=2} d(k)/prime(k) appears to converge, where d(k)=-1 if p mod 3 = 1, d(k)=+1 if p mod 3 = 2 and d(k)=0 if p mod 3 = 0.
Decimal expansion of the sum of 1/(p-1)^2 over all primes p.
Decimal expansion of Feller’s alpha coin-tossing constant.
Decimal expansion of Shapiro’s cyclic sum constant mu.
Decimal expansion of constant appearing in the expected number of comparisons for a successful digital tree search (negated).
Decimal expansion of constant appearing in the expected number of comparisons for an unsuccessful digital tree search (negated).
Decimal expansion of constant appearing in the variance for searching in a digital tree.
Decimal expansion of constant appearing in the variance for inserting in a digital tree.
Decimal expansion of constant c appearing in the expected number of pair of twin vacancies in a digital tree.
Decimal expansion of constant theta appearing in the expected number of pair of twin vacancies in a digital tree.
Decimal expansion of Lévy’s constant.
Decimal expansion of G(1/2) where G is the Barnes G-function.
Decimal expansion of 1/sqrt(Pi).
Decimal expansion of Krivine’s bound for Grothendieck’s constant, Pi/(2*log(1+sqrt(2))).
Decimal expansion of 6/(Pi^2 A086724).
Decimal expansion of Madelung’s constant M2.
Decimal expansion of 4/Pi.
Decimal expansion of (4K/Pi)^2 where K is the Landau-Ramanujan constant.
Decimal expansion of (4/sqrt(Pi))*exp(-gamma/2)*K where K is the Landau-Ramanujan constant and gamma the Euler-Mascheroni constant.
Decimal expansion of sqrt(Pi)/(2K)*exp(-gamma/2) where K is the Landau-Ramanujan constant and gamma the Euler-Mascheroni constant.
Decimal expansion of the analog of the Mertens constant B_2 in the asymptotic series for the variance of the number of prime factors Omega.
Decimal expansion of e^(-4).
Decimal expansion of 97/150 + Pi/40.
Decimal expansion of ”lemniscate case”.
Decimal expansion of constant in Kac’s formula.
Decimal expansion of sqrt(2)/phi, where phi = (1+sqrt(5))/2.
Decimal expansion of (4*sqrt(2)*Pi^2)/gamma(1/4)^2.
Decimal expansion of constant A*B in the asymptotic expression of the summatory function Sum_{n=1..N} (1/phi(n)) as A(log(N)+B) + O(log(N)/N).
Decimal expansion of the Paris constant.
Decimal expansion of the solution to zeta(x) = 2.
Decimal expansion of quadratic recurrence constant sqrt(1 * sqrt(2 * sqrt(3 * sqrt(4 * ...)))).
Decimal expansion of de Bruijn’s constant.
Decimal expansion of first zero of the Bessel function J_0(z).
Decimal expansion of (Glaisher^12/(2^(4/3) * Pi * e^EulerGamma))^(Pi^2/8).
Decimal expansion of (Glaisher^12/(2*Pi*e^EulerGamma))^(Pi^2/6).
Decimal expansion of 2 + 2*cos(2*Pi/7).
Coefficients in asymptotic expansion of sequence A052129.
Numbers of carefree couples (a,b) with a,b<=n.
Numbers of strongly carefree couples (a,b) with a,b <= n.
Number of weakly carefree couples (a,b) with a,b<=n.
Decimal expansion of probability of a weakly carefree couple.
Decimal expansion of the Embree-Trefethen constant.
A cubic recurrence: a(0) = 1, a(n) = n*a(n-1)^3 for n >= 1.
Decimal expansion of (1*(2*(3*...)^(1/3))^(1/3))^(1/3).
Numerators in an asymptotic expansion for the cubic recurrence sequence A123851.
Denominators in an asymptotic expansion for the cubic recurrence sequence A123851.
Decimal expansion of 2*exp(-gamma).
Decimal expansion of negative of Granville-Soundararajan constant.
Decimal expansion of the constant Sum_{k>=1} log(k + 1) / (k * (k + 1)).
Decimal expansion of 1/(2 log 2).
Decimal expansion of Sum_{p prime} 1/(p*(p-1)).
Decimal expansion of volume of the Meissner Body.
Decimal expansion of the Pell constant.
Decimal expansion of the Zolotarev-Schur constant.
Decimal expansion of the Flajolet-Odlyzko constant.
Decimal expansion of the Goh-Schmutz constant.
Decimal expansion of the Hall-Montgomery constant.
Decimal expansion of the Vallée constant.
Decimal expansion of Norton’s constant.
Decimal expansion of van der Corput’s constant.
Decimal expansion of the Takeuchi-Prellberg constant.
Decimal expansion of the paper-folding constant, or the dragon constant.
Decimal expansion of Sum_{k>=0} (k!/(k+2)!)^2.
Decimal expansion of 1 - gamma, where gamma is Euler’s constant (or the Euler-Mascheroni constant).
Decimal expansion of the Littlewood-Salem-Izumi constant.
Define a sequence of fractions by f(1) = 1/2, f(n+1) = (f(n)^2 + 1)/2; sequence gives numerators.
Decimal expansion of Selberg-Delange constant Product_{prime p > 2} (1 + 1/(p(p-2)))
Decimal expansion of (5 + 3*sqrt(5))/10.
Decimal expansion of Born’s basic potential Pi_0.
Decimal expansion of 1 - 1/Pi.
Decimal expansion of sqrt(2) - 1.
Decimal expansion of (diagonal)/(shortest side) of 2nd electrum rectangle.
Decimal expansion of the solution to x = sin( Pi/6 - x*sqrt(1 - x^2) ).
Decimal expansion of alpha, the unique solution on [2,oo) of the equation alpha*log((2*e)/alpha)=1.
Decimal expansion of the least x > 0 satisfying x + tan(x) = 0.
Decimal expansion of least x>0 having sin(4x) = (sin x)^2.
Perfect powers (squares, cubes, etc.) plus 1.
Decimal expansion of the generalized Stirling constant.
Decimal expansion of exp(gamma)/2.
Decimal expansion of the coefficient of asymptotic expression of m(n), the number of multiplicative compositions of n.
Decimal expansion of (3*sqrt(3)+sqrt(7))/10.
Decimal expansion of Baxter’s four-coloring constant.
Decimal expansion of Integral_{x=0..Pi/2} sin(x)^(3/2) dx.
Decimal expansion of Sum_{n>=0} 1/10^(3^n), a transcendental number.
Decimal expansion of the logarithm of Glaisher’s constant.
Decimal expansion of the 1st Lebesgue constant L1.
Decimal expansion of the 2nd Lebesgue constant L2.
Decimal expansion of the 3rd Lebesgue constant L3.
Second-order term in the asymptotic expansion of B(x), the count of numbers up to x which are the sum of two squares.
Decimal expansion of a Young-Fejér-Jackson constant linked to the nonnegativity of certain cosine sums.
Decimal expansion of ’lambda’, a Young-Fejér-Jackson constant linked to the positivity of certain sine sums.
Decimal expansion of ’mu’, a Young-Fejér-Jackson constant linked to the positivity of certain sine sums.
Decimal expansion of ’B’, a Young-Fejér-Jackson constant linked to the positivity of certain sine sums.
Decimal expansion of the side of the equilateral triangle that can cover every triangle of perimeter 2.
Decimal expansion of the integral_{x=0..Infinity} 1/x^x dx.
Decimal expansion of constant 1 + B3 (or 1 + B_3) related to the Mertens constant.
Numerator of 2n/v(n)^2, where v(1) = 0, v(2) = 1, and v(n) = v(n-1)/(n-2) + v(n-2) for n >= 3; limit of 2n/v(n)^2 is Pi.
Denominator of 2n/v(n)^2, where v(1) = 0, v(2) = 1, and v(n) = v(n-1)/(n-2) + v(n-2) for n >= 3. (Limit of 2n/v(n)^2 is Pi.)
Decimal expansion of 1/sqrt(2*Pi*e), one of the Traveling Salesman constants.
Decimal expansion of the radius of convergence of Wedderburn-Etherington numbers g.f.
Decimal expansion of the breadth of the ”caliper”, the broadest worm of unit length.
Decimal expansion of gamma’, the analog of Euler’s constant when 1/x is replaced by 1/(x*log(x)).
Decimal expansion of Sierpiński’s S^ (Ŝ or ”S hat” as named by S. Finch), a constant appearing in the asymptotics of the number of representations of a positive integer as a sum of two squares.
Decimal expansion of Sierpiński’s S  (S ”tilde” as named by S. Finch), a constant appearing in the asymptotics of the number of representations of a positive integer as a sum of two squares.
Decimal expansion of Sierpiński’s S constant, which appears in a series involving the function r(n), defined as the number of representations of the positive integer n as a sum of two squares. This S constant is the usual Sierpiński K constant divided by Pi.
Decimal expansion of an infinite product involving the ratio of n! to its Stirling approximation.
Numerator of phi(p-1)/(p-1), where phi is Euler’s totient function and p = prime(n).
Decimal expansion of D(1/2), where D(x) is the infinite product function defined in the formula section (or in the Finch reference).
Decimal expansion of D(1), where D(x) is the infinite product function defined in the formula section (or in the Finch reference).
Decimal expansion of delta = (1+alpha)/4, a constant appearing in Koecher’s formula for Euler’s gamma constant, where alpha is A065442, the Erdős-Borwein Constant.
Decimal expansion of the Euler-Kronecker constant (as named by P. Moree) for hypotenuse numbers.
Decimal expansion of the Euler-Kronecker constant (as named by P. Moree) for non-hypotenuse numbers.
Decimal expansion of 1/log(2)-1, the mean value of a random variable following the Gauss-Kuzmin distribution.
Decimal expansion of c, a constant appearing in the asymptotic lower bound of the size of a restricted difference set.
Decimal expansion of 2*Pi*phi(0), a constant appearing in connection with a study of zeros of the integral of xi(z), where phi(t) and xi(z) are functions related to Riemann’s zeta function (see Finch reference for the definition of these functions).
Decimal expansion of ’beta’, a constant appearing in the random links Traveling Salesman Problem.
Decimal expansion of the smallest positive root of the equation J_0(t)*I_1(t)+I_0(t)*J_1(t) = 0 (with I_0, I_1, J_0 and J_1, Bessel functions).
Decimal expansion of the binary self-numbers density constant.
Decimal expansion of the unforgeable pattern-free binary word constant, a constant mentioned in A003000.
Decimal expansion of one of the Pell-Stevenhagen constants.
Decimal expansion of the coefficient C used in the asymptotic evaluation of the number of primitive Pythagorean triangles with area less than n, as C*sqrt(n).
Decimal expansion of a constant related to a certain Sobolev isoperimetric inequality.
Decimal expansion of the first positive solution to exp(1-1/x)/x = 1/2, a binary search tree constant.
a(n) is the number of not-sqrt-smooth numbers (”jagged” numbers) not exceeding n. This is the counting function of A064052.
Decimal expansion of Sum_{k > 0} (-1)^k*log(k)^2/k.
Decimal expansion of the expected reciprocal Euclidean distance between two random points in the unit cube.
Decimal expansion of 1-gamma-gamma(1), a constant related to the asymptotic expansion of j(n), the counting function of ”jagged” numbers, where gamma is Euler-Mascheroni constant and gamma(1) the first Stieltjes constant.
Decimal expansion of the sum of the alternating series tau(3), with tau(n) = Sum_{k>0} (-1)^k*log(k)^n/k.
Decimal expansion of the sum of the alternating series tau(4), with tau(n) = Sum_{k>0} (-1)^k*log(k)^n/k.
Decimal expansion of the sum of the alternating series tau(5), with tau(n) = Sum_{k>0} (-1)^k*log(k)^n/k.
Decimal expansion of the (real) period of the elliptic function sn(x,1/2).
Decimal expansion of k2, a Diophantine approximation constant such that the area of the ”critical parallelogram” (in this case a square) is 4*k2.
Decimal expansion of an optimal stopping constant related to the Secretary problem.
Decimal expansion of the unique real solution of the equation Ei(x)-gamma-log(x) = 1, where Ei is the exponential integral function and gamma the Euler-Mascheroni constant.
Decimal expansion of the asymptotic probability of success in one of the Secretary problems.
Decimal expansion of the positive real root of 3*x^4 - x^3 - x^2 - 2, a constant related to quasi-isometric mappings.
Decimal expansion of a constant associated with self-generating continued fractions and Cahen’s constant.
Decimal expansion of the Atkinson-Negro-Santoro constant, a constant associated with Erdős’ sum-distinct set constant.
Decimal expansion of the Conway-Guy constant, a constant associated with Erdős’ sum-distinct set constant.
Decimal expansion of an Ising constant related to the random coloring problem.
Decimal expansion of the even limit of the harmonic power tower (1/2)^(1/3)^...^(1/(2n)).
Decimal expansion of the odd limit of the harmonic power tower (1/2)^(1/3)^...^(1/(2n+1)).
Decimal expansion of the escape probability for a random walk on the 3-D cubic lattice (a Polya random walk constant).
Decimal expansion of the positive solution to the equation x/(1-x) = 1+log(1/(1-x)), an auxiliary constant associated with the problem of enumeration of trees by inversions.
Decimal expansion of the maximum probability that the convex hull of four points, chosen at random inside a convex planar region, is a quadrilateral (Sylvester’s four-point problem).
Decimal expansion of the expected number of returns to the origin of a random walk on a 4-d lattice.
Decimal expansion of the expected number of returns to the origin of a random walk on a 5-d lattice.
Decimal expansion of the expected number of returns to the origin of a random walk on a 6-d lattice.
Decimal expansion of the expected number of returns to the origin of a random walk on a 7-d lattice.
Decimal expansion of the expected number of returns to the origin of a random walk on an 8-d lattice.
Decimal expansion of B. Davis’ constant Pi^2/(8*G), a Riesz-Kolmogorov constant, where G is Catalan’s constant.
Decimal expansion of exp(7*zeta(3)/(2*Pi^2)).
Decimal expansion of exp(-gamma/2).
Decimal expansion of exp(sqrt(Pi/24)).
Decimal expansion of the trace of the Ruelle-Mayer linear operator G_2.
Decimal expansion of c_e, coefficient associated with the asymptotic evaluation c_e*2^(n^2/4) of the number of subspaces of the n-dimensional vector space over the finite field F_2, n being even.
Decimal expansion of c_o, coefficient associated with the asymptotic evaluation c_o*2^(n^2/4) of the number of subspaces of the n-dimensional vector space over the finite field F_2, n being odd.
Decimal expansion of the mean car density associated with Solomon’s variation in Renyi’s one-dimensional parking problem.
Decimal expansion of H, an auxiliary constant used to evaluate some Ising-related constants on triangular and hexagonal lattices (negated).
Decimal expansion of an Ising constant related to the triangular lattice.
Decimal expansion of an Ising constant related to the hexagonal lattice.
Decimal expansion of a constant related to Niven’s constant.
Decimal expansion of Sum_{k>1} 1/(k*(k-1)*zeta(k)), a constant related to Niven’s constant.
Decimal expansion of a constant related to the asymptotic expansion of the Lebesgue constant corresponding to the n-th Chebyshev polynomial.
Decimal expansion of a constant related to the asymptotic expansion of the smallest Lebesgue constant corresponding to an optimal interpolation data set.
Decimal expansion of a Shapiro-Drinfeld constant, known as Gauchman’s constant, related to the difference of cyclic sums (negated).
Decimal expansion of the generalized Glaisher-Kinkelin constant A(2).
Decimal expansion of the generalized Glaisher-Kinkelin constant A(3).
Decimal expansion of the generalized Glaisher-Kinkelin constant A(4).
Decimal expansion of the generalized Glaisher-Kinkelin constant A(5).
Decimal expansion of a parking constant related to the asymptotic expected number of cars, assuming random car lengths.
Decimal expansion of 1/2+G/Pi, the highest limiting crest of a square wave Fourier series, where G is the Gibbs-Wilbraham constant.
Decimal expansion of 1/2-G/Pi, the lowest limiting trough of a square wave Fourier series, where G is the Gibbs-Wilbraham constant. [negated]
Decimal expansion of ’c’, a constant related to the asymptotic evaluation of the Lebesgue constants L_n.
Decimal expansion of a constant related to the asymptotic evaluation of the Lebesgue constants L_n.
Decimal expansion of h_3, a constant related to certain evaluations of the gamma function from elliptic integrals.
Decimal expansion of DeVicci’s tesseract constant.
Decimal expansion of the volume of a regular ideal hyperbolic 4-simplex.
Decimal expansion of a 5-dimensional analog of DeVicci’s tesseract constant.
Decimal expansion of sqrt(8/Pi)*log(2), a constant related to the asymptotic evaluation of the minimum number of one-dimensional random walks that have to be examined to compute the maximum.
Decimal expansion of 4*L/(3*Pi), a constant related to the asymptotic evaluation of the number of primes of the form a^2+b^4, where L is Gauss’ lemniscate constant.
Decimal expansion of the unique solution of the equation sum_(p prime)(1/p^x) = 1, a constant related to the asymptotic evaluation of the number of prime multiplicative compositions.
Decimal expansion of 6*K/Pi^2, a constant related to the asymptotic evaluation of the number of positive squarefree integers of the form a^2 + b^2, where K is the Landau-Ramanujan constant.
Decimal expansion of 3/(8*K), a constant related to the asymptotic evaluation of the number of positive integers that can be expressed as the sum of two coprime squares, where K is the Landau-Ramanujan constant.
Decimal expansion of 1/(4*K), a constant related to the asymptotic evaluation of the number of positive integers all of whose prime factors are congruent to 1 modulo 4, where K is the Landau-Ramanujan constant.
Decimal expansion of 2*K/Pi, a constant related to the asymptotic evaluation of the number of positive integers all of whose prime factors are congruent to 3 modulo 4, where K is the Landau-Ramanujan constant.
Decimal expansion of a constant related to the asymptotic evaluation of Product_{p prime congruent to 1 modulo 4} (1 + 1/p).
Decimal expansion of a constant related to the asymptotic evaluation of Product_{p prime congruent to 3 modulo 4} (1 + 1/p).
Decimal expansion of 1/(2*K^2) = Product_(p prime congruent to 3 modulo 4) (1 - 1/p^2), where K is the Landau-Ramanujan constant.
Decimal expansion of 192*K^2*G/Pi^4 = Product_{p prime congruent to 1 modulo 4} (1 + 1/p^2), where K is the Landau-Ramanujan constant and G Catalan’s constant.
Decimal expansion of Pi^2/(16*K^2*G) = Product_{p prime congruent to 3 modulo 4} (1 + 1/p^2), where K is the Landau-Ramanujan constant and G Catalan’s constant.
Decimal expansion of ’xiHat’, a constant used in the asymptotic evaluation of e.g.f. coefficients for the number of labeled mobiles.
Decimal expansion of ’etaHat’, a constant used in the asymptotic evaluation of e.g.f. coefficients for the number of labeled mobiles.
Decimal expansion of the limiting length appearing in the asymptotic probability involved in the ”stick breaking” problem.
Decimal expansion of c = twice the maximum of Dawson’s integral, a constant used in the asymptotic evaluation of the ideal hyperbolic n-cube volume.
Decimal expansion of c*sqrt(e/2), a constant associated with Dawson’s integral and the asymptotic evaluation of the ideal hyperbolic n-cube volume, where c is A243433, twice the maximum of Dawson’s integral.
Decimal expansion of 3/(2*sqrt(Pi)).
Decimal expansion of 1-9/(4*Pi)+sqrt(3)/(2*Pi), an extreme value constant.
Decimal expansion of 6*arcsec(sqrt(3))/Pi^(3/2), an extreme value constant.
Decimal expansion of the variance of the maximum of a size 4 sample from a normal (0,1) distribution.
Decimal expansion of the expectation of the maximum of a size 5 sample from a normal (0,1) distribution.
Decimal expansion of the variance of the maximum of a size 5 sample from a normal (0,1) distribution.
Decimal expansion of the expectation of the maximum of a size 6 sample from a normal (0,1) distribution.
Decimal expansion of the expectation of the maximum of a size 7 sample from a normal (0,1) distribution.
Decimal expansion of the variance of the maximum of a size 6 sample from a normal (0,1) distribution.
Decimal expansion of the variance of the maximum of a size 7 sample from a normal (0,1) distribution.
Decimal expansion of ’c’, an asymptotic constant related to a variation of the ”Secretary problem” with a uniform distribution.
Decimal expansion of 1/(eta*P’(eta)), a constant related to the asymptotic evaluation of the number of prime multiplicative compositions, where eta is A243350, the unique solution of P(x)=1, P being the prime zeta P function (P(x) = sum_(p prime) 1/p^x).
Decimal expansion of the expectation of the maximum of a size 8 sample from a normal (0,1) distribution.
Decimal expansion of the variance of the maximum of a size 8 sample from a normal (0,1) distribution.
Decimal expansion of ’xi’, a constant related to the second order quadratic recurrence q(0)=q(1)=1, q(n)=q(n-2)*(q(n-1)+1).
Decimal expansion of ’eta’, a constant related to the second order quadratic recurrence q(0)=q(1)=1, q(n)=q(n-2)*(q(n-1)+1).
Decimal expansion of B (negated), a constant related to Glaisher’s constant A and the Gaussian unitary ensemble hypothesis.
Decimal expansion of the Bateman-Grosswald constant zeta(2/3)/zeta(2), a constant (negated) arising in the asymptotic evaluation of the number of square-full numbers (also called ”powerful” numbers).
Decimal expansion of the Greenfield-Nussbaum constant, a constant which is the term z(1) in the quadratic recurrence z(0)=1, z(n) = z(n-1)+z(n-2)^2, such that all terms of the bi-infinite sequence z(n) (n = ..., -2, -1, 0, 1, 2, ...) are positive.
Decimal expansion of the maximal width of a Reuleaux triangle avoiding all vertices of the integer square lattice.
Decimal expansion of the maximal circumradius of a planar convex set containing no lattice point except for the origin where it has its circumcenter, and not protruding in opposite directions outside the square max(|x|,|y|) < 1 unless it protrudes significantly elsewhere, too.
Decimal expansion of the Purdom-Williams constant, a constant related to the Golomb-Dickman constant and to the asymptotic evaluation of the expectation of a random function longest cycle length.
Decimal expansion of 1/2+2/sqrt(13), a constant related to the asymptotic evaluation of the number of self-avoiding rook paths joining opposite corners on a 3 X n chessboard.
Decimal expansion of sqrt((3+sqrt(13))/2), a constant related to the asymptotic evaluation of the number of self-avoiding rook paths joining opposite corners on a 3 X n chessboard.
Decimal expansion of sqrt(3)/(2*(sqrt(2)-1))^(1/3), the Landau-Kolmogorov constant C(3,1) for derivatives in L_2(0, infinity).
Decimal expansion of a partial sum limiting constant related to the Lüroth representation of real numbers.
Decimal expansion of K = exp(8*G/(3*Pi)), a Kneser-Mahler constant related to an asymptotic inequality involving Bombieri’s supremum norm, where G is Catalan’s constant. K can be evaluated as Mahler’s generalized height measure of the bivariate polynomial (1+x+x^2+y)^2.
Decimal expansion of exp(gamma)/sqrt(2)*Product_{n>=1} ((2n+1)/(2n))^((-1)^t(n)), a probabilistic counting constant, where gamma is Euler’s constant and t(n) = A010060(n) the Thue-Morse sequence.
Decimal expansion of the asymptotic evaluation of the constrained maximum of a certain quadratic form.
Decimal expansion of sqrt(2*Pi)*log(2), a constant associated with asymptotic evaluation of random mapping statistics.
Decimal expansion of c = 2.4149..., a random mapping statistics constant such that the asymptotic expectation of the maximum rho length (graph diameter) in a random n-mapping is c*sqrt(n).
Decimal expansion of theta = 2.472548..., an auxiliary constant used to compute the best constant in Friedrichs’ inequality in one dimension.
Decimal expansion of beta = 1.07869..., the best constant in Friedrichs’ inequality in one dimension.
Decimal expansion of exp(gamma)/log(2), a conjectural constant related to the asymptotic counting of Mersenne primes, where gamma is Euler’s constant.
Decimal expansion of A1*B1, the average number of non-isomorphic semisimple rings of any order, where A1 is Product_{m>1} zeta(m) and B1 is Product_{r*m^2 > 1} zeta(r*m^2).
Decimal expansion of B, a constant related to the Goh-Schmutz constant and the asymptotic expected order of a random permutation.
Decimal expansion of 3/2 - gamma / log(2), a coin tossing constant related to the asymptotic evaluation of the expected length of the longest run of consecutive heads.
Decimal expansion of 64/169, the upper bound (as given by S. Finch) of the 2-dimensional simultaneous Diophantine approximation constant.
Decimal expansion of 1/(2*(Pi-2)), the upper bound of the 3-dimensional simultaneous Diophantine approximation constant.
Decimal expansion of the upper bound of the 4-dimensional simultaneous Diophantine approximation constant.
Decimal expansion of the upper bound of the 5-dimensional simultaneous Diophantine approximation constant.
Decimal expansion of the upper bound of the 6-dimensional simultaneous Diophantine approximation constant.
Decimal expansion of 56/13, the Korn constant for the sphere.
Decimal expansion of ’mu’, a Sobolev isoperimetric constant related to the ”rod inequality”, arising from the elasticity study of a rod that is clamped at both ends.
Decimal expansion of ’lambda’, a Sobolev isoperimetric constant related to the ”rod inequality”, arising from the elasticity study of a rod that is clamped at both ends.
Decimal expansion of ’mu’, a Sobolev isoperimetric constant related to the ”membrane inequality”, arising from the study of a vibrating membrane that is stretched across the unit disk and fastened at its boundary.
Decimal expansion of ’lambda’, a Sobolev isoperimetric constant related to the ”membrane inequality”, arising from the study of a vibrating membrane that is stretched across the unit disk and fastened at its boundary.
Decimal expansion of ’lambda’, a constant such that exp(lambda*Pi) is the best-known upper bound (as given by Julian Gevirtz) of the John constant for the unit disk.
Decimal expansion of the best-known upper bound (as given by Julian Gevirtz) of the John constant for the unit disk.
Decimal expansion of z_c = phi^5 (where phi is the golden ratio), a lattice statistics constant which is the exact value of the critical activity of the hard hexagon model.
Decimal expansion of the coefficient D appearing in the asymptotic evaluation of P_a(n), the number of primitive Pythagorean triples whose area does not exceed a given bound n.
Decimal expansion of ’theta’, the unique positive root of the equation polygamma(x) = log(Pi), where polygamma(x) gives gamma’(x)/gamma(x), that is the logarithmic derivative of the gamma function.
Decimal expansion of 4*K/Pi, a constant appearing in the asymptotic evaluation of the number of non-hypotenuse numbers not exceeding a given bound, where K is the Landau-Ramanujan constant.
Decimal expansion of ’C’ (as designated by D. Shanks), a constant appearing in the second order term of the asymptotic expansion of the number of non-hypotenuse numbers not exceeding a given bound.
Decimal expansion of Ti_2(2-sqrt(3)), where Ti_2 is the inverse tangent integral function.
Decimal expansion of Ti_2(2+sqrt(3)), where Ti_2 is the inverse tangent integral function.
Decimal expansion of the moment derivative W_3’(0) associated with the radial probability distribution of a 3-step uniform random walk.
Decimal expansion of ’tau’ (named sigma_2 by C. Pomerance), a constant associated with the expected number of random elements to generate a finite abelian group.
Number of three-dimensional random walks with 2n steps in the wedge region x >= y >= z, beginning and ending at the origin without crossing the wedge boundary.
Decimal expansion of Integral_{x=0..Pi/2} (x^2/sin(x)) dx.
Decimal expansion of B, the coefficient of n*log(n)^2 in the asymptotic formula of Ramanujan for Sum_{k=1..n} (d(k)^2), where d(k) is the number of distinct divisors of k.
Decimal expansion of the Landau-Kolmogorov constant C(3,1) for derivatives in the case L_infinity(-infinity, infinity).
Decimal expansion of the Landau-Kolmogorov constant C(3,1) for derivatives in the case L_infinity(0, infinity).
Decimal expansion of the average value of the Yekutieli-Mandelbrot parameter, that is the average number of maximal subtrees of an ordered binary tree requiring one less register than the whole tree.
Decimal expansion of d_0, the constant term in the asymptotic expansion of the average number of registers needed to evaluate a binary tree.
Decimal expansion of U = Product_{k>=1} (k^(1/(k*(k+1)))), a Khintchine-like limiting constant related to Lüroth’s representation of real numbers.
Decimal expansion of y_1, the first of four non-explicit constants recursively derived from Khintchine’s [Khinchin’s] constant.
Decimal expansion of y_2, the second of four non-explicit constants recursively derived from Khintchine’s [Khinchin’s] constant.
Decimal expansion of y_3, the third of four non-explicit constants recursively derived from Khintchine’s [Khinchin’s] constant.
Decimal expansion of y_4, the last of four non-explicit constants recursively derived from Khintchine’s [Khinchin’s] constant.
Decimal expansion of Dawson’s integral at the inflection point.
Decimal expansion of zeta”(0)+2, the coefficient of z^2 in the Laurent expansion of zeta(z) at the origin.
Decimal expansion of sum_{r in Z}(1/r^2) where Z is the set of all nontrivial zeros r of the Riemann zeta function.
Decimal expansion of sum_{r in Z}(1/r^3) where Z is the set of all nontrivial zeros r of the Riemann zeta function.
Decimal expansion of k3, a Diophantine approximation constant such that the conjectured volume of the ”critical parallelepiped” is 2^3*k3 (the 3-D analog of A242671).
Decimal expansion of a1, the first of two constants associated with Djokovic’s conjecture on an integral inequality.
Decimal expansion of a2, the second of two constants associated with Djokovic’s conjecture on an integral inequality.
Decimal expansion of the Landau-Kolmogorov constant C(4,1) for derivatives in L_2(0, infinity).
Decimal expansion of the Landau-Kolmogorov constant C(4,2) for derivatives in L_2(0, infinity).
Decimal expansion of ’mu’, an isoperimetric constant associated with the study of a vibrating, homogeneous plate clamped at the boundary of the unit disk.
Decimal expansion of the Landau-Kolmogorov constant C(4,1) for derivatives in the case L_infinity(infinity, infinity).
Decimal expansion of the square root of 6/5.
Decimal expansion of the Landau-Kolmogorov constant C(4,3) for derivatives in the case L_infinity(infinity, infinity).
Decimal expansion of the Landau-Kolmogorov constant C(5,1) for derivatives in the case L_infinity(infinity, infinity).
Decimal expansion of the Landau-Kolmogorov constant C(5,2) for derivatives in the case L_infinity(infinity, infinity).
Decimal expansion of the Landau-Kolmogorov constant C(5,3) for derivatives in the case L_infinity(infinity, infinity).
Decimal expansion of the Landau-Kolmogorov constant C(5,4) for derivatives in the case L_infinity(-infinity, infinity).
Decimal expansion of c_1, a constant associated with the computation of the maximal modulus of an algebraic integer.
Decimal expansion of phi(0), an auxiliary constant associated with Shapiro’s cyclic sum constant lambda.
Decimal expansion of the 6th du Bois-Reymond constant.
Decimal expansion of b_3, a constant associated with the 3rd Du Bois Reymond constant.
Decimal expansion of sigma_3, a Turán’s Power Sum Constant.
Decimal expansion of the analog of the Gibbs-Wilbraham constant for L_1 trigonometric polynomial approximation.
Decimal expansion of the Ising constant K_c, the ratio of the coupling constant to the ferromagnetic critical temperature, in the two-dimensional case.
Decimal expansion of ’chi’, a constant appearing in the asymptotic variance of the number of comparisons required for updating a digital search tree, in case of the ”approximate counting” algorithm.
Decimal expansion of Integral_{x = 1..infinity} 1/x^x dx.
Decimal expansion of the lower bound of the Berry-Esseen constant.
Decimal expansion of eta/xi = A086318/A086317, a coefficient associated with the asymptotics of the number of weakly binary trees.
Decimal expansion of a coefficient associated with the asymptotics of the average distance between a vertex and the root of a random rooted tree.
Decimal expansion of a coefficient associated with the asymptotics of the variance of the distance between a vertex and the root of a random rooted tree.
Decimal expansion of the smallest positive root of the function lambda(x) = sum_{n=0..infinity} (-1)^n*x^n/(2^(n*(n-1)/2)*n!).
Decimal expansion of eta_A, a constant associated with the asymptotics of the enumeration of labeled acyclic digraphs.
Decimal expansion of k_3 = 3/(2*Pi*m_3), a constant associated with the asymptotic expansion of the probability that a three-dimensional random walk reaches a given point for the first time, where m_3 is A086231 (Watson’s integral).
Decimal expansion of C, a coin tossing constant related to the asymptotic variance of the length of the longest run of consecutive heads.
Decimal expansion of ’nu’, a coefficient related to the variance for searching corresponding to patricia tries.
Decimal expansion of the skewness of the Gumbel distribution.
Decimal expansion of the mean cluster density for bond percolation on the triangular lattice.
Decimal expansion of ’g’, a constant related to the asymptotic distribution of the Q moment and the logarithmic divergence of the Ising specific heat.
Decimal expansion of z_tri, a constant related to the enumeration of spanning trees on the triangular lattice (this is different from A242968).
Decimal expansion of ’b’, an optimal stopping constant associated with the secretary problem when the objective is to maximize the hiree’s expected quality.
Decimal expansion of the entropy of folding of the triangular lattice.
Decimal expansion of (1-C_2)/e, a constant connected with two-sided generalized Fibonacci sequences, where C_2 is the Euler-Gompertz constant.
Decimal expansion of the infinite product of (j/Pi)*sin(Pi/j), for j >= 2, a constant similar to the Kepler-Bouwkamp constant.
Decimal expansion of lim_{n->infinity} ((1/log(n)^2)*Product_{2 < p < n, p prime} p/(p-2)).
Decimal expansion of Hermite’s constant gamma_6 = 2/3^(1/6).
Decimal expansion of ’a’, an auxiliary constant associated with the asymptotic probability of success in the secretary problem when the number of applicants is uniformly distributed.
Decimal expansion of the asymptotic probability of success in the full-information secretary problem with uniform distribution when the number of applicants is also uniformly distributed.
Decimal expansion of ’b’, an auxiliary constant associated with the asymptotic probability of success in the full information version of the secretary problem.
Decimal expansion of the asymptotic probability of success in the full information version of the secretary problem.
Decimal expansion of Shepp’s constant ’alpha’, an optimal stopping constant associated with the case of a zero mean and unit variance distribution function.
Decimal expansion of ’xi’, an optimal stopping auxiliary constant associated with the two choice case.
Decimal expansion of ’c’, an optimal stopping constant associated with the two choice case.
Decimal expansion of ’mu’, a percolation constant associated with the asymptotic threshold for 3-dimensional bootstrap percolation.
Decimal expansion of integral_{0..infinity} x*log(x)*(1-eta(x)^2) dx, where the function ’eta’ is a solution of the Painlevé III differential equation.
Decimal expansion of Hermite’s constant gamma_7 = 2^(6/7).
Decimal expansion of a limit associated with the asymptotic number of ways of writing a number as a sum of powers of 2, with each power used at most twice (cardinality of ”alternating bit sets” of a given number, also known as Stern’s diatomic sequence).
Decimal expansion of the best constant K for the integral inequality integral_{0..1} f(x)^2*f’(x)^2 dx <= K*integral_{0..1} f’(x)^4 dx.
Decimal expansion of H_2, the analog of Madelung’s constant for the planar hexagonal lattice.
Decimal expansion of integral_{0..infinity} exp(-x^2)*log(x) dx.
Decimal expansion of 2*G/(Pi*log(2)), a constant appearing in the average root bifurcation ratio of binary trees, where G is Catalan’s constant.
Decimal expansion of Sum_{k >= 0} 1/(4*k+3)^2.
Decimal expansion of Integral_{x=0..1} log(floor(1/x))/(1+x) dx.
Decimal expansion of the analog of Lévy’s constant in case of the nearest integer continued fraction of -1/2<x<1/2.
Decimal expansion of M_6, the 6th Madelung constant.
Decimal expansion of zeta(5/2).
Decimal expansion of delta_2 (negated), a constant associated with a certain two-dimensional lattice sum.
Decimal expansion of gamma_2, a lattice sum constant, analog of Euler’s constant for two-dimensional lattices.
Decimal expansion of delta_3, a constant associated with a certain 3-dimensional lattice sum.
Decimal expansion of lambda_3, an analog of the Stolarsky-Harborth constant for the number of elements not divisible by 3 in Pascal’s triangle.
Decimal expansion of gamma_3, a lattice sum constant, analog of Euler’s constant for 3-dimensional lattices.
Decimal expansion of C = 2^(1/3)*e^(1/4)/A^3, a constant associated with the Gaudin-Mehta probability distribution and the Glaisher-Kinkelin constant A.
Decimal expansion of p_2, a probability associated with continuant polynomials.
Least m such that (4/m^2)*Sum_{k=0..m} sqrt(m^2 - k^2) < Pi + 1/n.
Decimal expansion of the Landau-Kolmogorov constant C(5,1) for derivatives in L_2(0, infinity).
Decimal expansion of Integral_{0..oo} 1/Gamma(1+x) dx, a variation of the Fransén-Robinson constant.
Decimal expansion of ’v’, a parking constant associated with the asymptotic variance of the number of cars that can be parked in a given interval.
Decimal expansion of a constant ’v’ such that the asymptotic variance of the distribution of the longest cycle given a random n-permutation evaluates as v*n^2.
Decimal expansion of A(rectangles), an analog of Moser’s worm constant, which is associated with the class of rectangular regions of the plane.
Decimal expansion of a(F_5), the maximum inradius of all triangles that lie in a regular pentagon of width 1.
Decimal expansion of the coefficient c_md in c_md*log(N)^(1/rho), the asymptotic mean number of distinct factors in a random factorization of n <= N.
Decimal expansion of the coefficient c_m in c_m*log(N), the asymptotic mean number of factors in a random factorization of n <= N.
Decimal expansion of the coefficient c_v in c_v*log(N), the asymptotic variance of the number of factors in a random factorization of n <= N.
Decimal expansion of the coefficient c appearing in the asymptotic evaluation of the number of prime additive compositions of n as c*(1/xi)^n, where xi is A084256.
Decimal expansion of 1/(theta*P’(theta)), a constant appearing in the asymptotic evaluation of the coefficients q_n in 1/(1+P(x)), where P(x) is the generating function of the primes and theta the unique zero of P(x) in [-3/4, 0].
Least number k such that u(k) - Pi < 1/5^n, where u is defined using the Borchardt-Pfaff algorithm; see Comments.
Least number k such that Pi - v(k) < 1/5^n, where v is defined using the Borchardt-Pfaff algorithm; see Comments.
Decimal expansion of m = (1-1/e^2)/2, one of Renyi’s parking constants.
Decimal expansion of m_2 = (2-1/e)/4, one of Renyi’s parking constants, the mean car density in case ”monomer with nearest neighbor exclusion” for the 2 x infinity strip.
Least number k such that e - 2*k/u(2*k) < 1/n^n, where u is defined as in Comments.
Least number k such that (2*k+1)/u(2*k+1) - e < 1/n^n, where u is defined as in Comments.
Least number k such that |(k+1)/u(k+1) - e| < 1/n^n, where u is defined as in Comments.
Decimal expansion of the value of a nonregular continued fraction giving tau/(3*tau-1), where tau is the Prouhet-Thue-Morse constant.
a(n) = least k such that (k!*e^k)/(sqrt(2*Pi)*k^(k+1/2)) - 1 < 1/2^n.
Least k such that 4*k/v(2*k)^2 - Pi < 1/n, where the sequence v is defined in Comments.
Least k such that Pi - (4*k+2)/v(2*k+2)^2 < 1/n, where the sequence v is defined in Comments.
Least number k such that log(2) - sum{1/(h*2^h), h=1..k} < 1/2^n.
Least number k such that product{(k^2 + h)/(k^2 - h), h = 1..k} - e < 1/n.
Least number k such that e - k/(k!)^(1/k) < 1/n.
Least k such that ((2k+1)/(2k-1))^k < 1/(2n^2).
Least k such that ((k+1)/(k-1))^k - e^2 < 1/n^2.
Least k such that ((k+2)/(k-2))^k - e^4 < 1/n.
a(n) = floor(1 / (1/n - Pi^2/6 + Sum_{h=1..n} 1/h^2)).
Decimal expansion of product_{n>=1} (n/(n+1))^((-1)^t(n)), a probabilistic counting constant, where t(n) = A010060(n) is the Thue-Morse sequence.
Decimal expansion of the best lower bound for the Steiner ratio rho_3, the least upper bound on the ratio of the length of the Steiner minimal tree to the length of the minimal tree in dimension 3.
Decimal expansion of theta_1, one of the angles associated with the bow-and-arrow configuration used in the 2-arc smallest length problem.
Decimal expansion of theta_2, one of the angles associated with the bow-and-arrow configuration used in the 2-arc smallest length problem.
Decimal expansion of the upper bound on length associated with the bow-and-arrow configuration used in the 2-arc smallest length problem.
Least k such that log(2) - sum{1/(h*2^h), h = 1..k} < 1/3^n.
Least k such that log(3/2) - sum{1/(h*3^h), h = 1..k} < 1/6^n.
Least k such that log(4/3) - sum{1/(h*4^h), h = 1..k} < 1/8^n.
Decimal expansion of product_{n>=1} (n/(n+1))^((-1)^t(n-1)), a probabilistic counting constant, where t(n) = A010060(n) is the Thue-Morse sequence.
Decimal expansion of product_{n>=1} (2n/(2n+1))^((-1)^t(n-1)), a probabilistic counting constant, where t(n) = A010060(n) is the Thue-Morse sequence.
Decimal expansion of I, a constant appearing (as I^2) in the asymptotic variance of the area of the convex hull of random points in the unit square.
a(n) = (2*n+23)*n*(n-1), a coefficient appearing in the formula a(n)*Pi/324+n+1 giving the average number of regions into which n random planes divide the cube.
Least k such that Pi/2 - sum{2^h/((2h+1)*C(2h,h)), h = 1..k} < 1/3^n.
Least k such that (Pi^2)/18 - sum{1/(h^2*C(2h,h)), h = 1..k} < 1/3^n.
Decimal expansion of a variant of the Komornik-Loreti constant.
Decimal expansion of a second variant of the Komornik-Loreti constant.
Decimal expansion of L = integral_{0..1} 1/(1-2t^2/3) dt, an auxiliary constant associated with one of the integral inequalities studied by David Boyd.
Decimal expansion of gamma = 8*lambda^2, a critical threshold of a boundary value problem, where lambda is Laplace’s limit constant A033259.
Decimal expansion of the Flajolet-Prodinger constant ’K’, a constant related to asymptotically enumerating level number sequences for binary trees.
Decimal expansion of ’sigma’, a constant associated with the expected number of random elements to generate a finite abelian group.
Decimal expansion of the logarithmic capacity of the unit disk.
Decimal expansion of the logarithmic capacity of the unit equilateral triangle.
Decimal expansion of a constant related to the [dimensionless] electrical capacitance of the ring torus without hole (with unit circle radius).
Decimal expansion of the mean value over all positive integers of a function giving the least quadratic nonresidue modulo a given odd integer (this function is precisely defined in A053761).
Numerators of 2*H(n)-H(n*(n+1)), a sequence the limit of which is gamma, the Euler-Mascheroni constant, where H(n) is the n-th harmonic number.
Denominators of 2*H(n)-H(n*(n+1)), a sequence the limit of which is gamma, the Euler-Mascheroni constant, where H(n) is the n-th harmonic number.
Product_{k=1..n} k^(k^5).
Decimal expansion of Mrs. Miniver’s constant.
Decimal expansion of sec(phi), a constant related to the ”broadworm” (or ”caliper”) problem.
Decimal expansion of zeta’(-3) (the derivative of Riemann’s zeta function at -3).
Decimal expansion of zeta’(-4) (the derivative of Riemann’s zeta function at -4).
Decimal expansion of zeta’(-5) (the derivative of Riemann’s zeta function at -5) (negated).
Decimal expansion of zeta’(-6) (the derivative of Riemann’s zeta function at -6) (negated).
Decimal expansion of zeta’(-7) (the derivative of Riemann’s zeta function at -7) (negated).
Decimal expansion of zeta’(-8) (the derivative of Riemann’s zeta function at -8).
Decimal expansion of m_3, the expected number of returns to the origin in a three-dimensional random walk restricted to the region x >= y >= z.
Decimal expansion of the constant c_0 appearing in the asymptotic evaluation of the n-th Lebesgue constant (related to Fourier series) as L_n   (4/Pi^2)*log(n) + c_0.
Decimal expansion of ’theta’, the expected degree (valency) of the root of a random rooted tree with n vertices.
Decimal expansion of the radius of convergence of the generating function of A000598, the number of rooted ternary trees of n vertices.
Decimal expansion of the Markoff number asymptotic density constant.
Decimal expansion of zeta(7/2).
Decimal expansion of M_8, the 8th Madelung constant (negated).
Decimal expansion of -Zeta’(-1)/2.
Decimal expansion of Integral_{0..1/2} log(gamma(x+1)) dx (negated).
Decimal expansion of H(1/2,1), a constant appearing in the asymptotic variance of the largest component of random mappings on n symbols, expressed as H(1/2,1)*n^2.
Decimal expansion of the coefficient ’gamma’ (see formula) appearing in Otter’s result concerning the asymptotics of T_n, the number of non-isomorphic rooted trees of order n.
Decimal expansion of C_{1/2}, a constant related to Kolmogorov’s inequalities.
Decimal expansion of M_5, the 5-dimensional analog of Madelung’s constant (negated).
Decimal expansion of M_7, the 7-dimensional analog of Madelung’s constant (negated).
Decimal expansion of Product_{p prime > 2} 1-1/(p^2-3p+3), a constant related to I. M. Vinogradov’s proof of the ”ternary” Goldbach conjecture.
Decimal expansion of ’lambda’, a Somos quadratic recurrence constant mentioned by Steven Finch.
Decimal expansion of Hardy-Littlewood constant C_5 = Product_{p prime > 5} 1/(1-1/p)^5 (1-5/p).
Decimal expansion of Hardy-Littlewood constant C_6 = Product_{p prime > 6} 1/(1-1/p)^6 (1-6/p).
Decimal expansion of Hardy-Littlewood constant C_7 = Product_{p prime > 7} 1/(1-1/p)^7 (1-7/p).
Decimal expansion of Product_{p odd prime} 1-2/(p*(p-1)), a constant related to Artin’s conjecture in the context of quadratic fields.
Decimal expansion of Matthews’ constant C_2, an analog of Artin’s constant for primitive roots.
Decimal expansion of Matthews’ constant C_3, an analog of Artin’s constant for primitive roots.
Decimal expansion of a constant related to the expected number of vertices of the largest tree associated with a random mapping on n symbols.
Decimal expansion of the doubly infinite sum N_3 = Sum_{i,j,k = -inf..inf} (-1)^(i+j+k)/(i^2+j^2+k^2), a lattice constant analog of Madelung’s constant (negated).
Decimal expansion of Matthews’ constant C_4, an analog of Artin’s constant for primitive roots.
Decimal expansion of the constant D related to the conjectured asymptotic expression of the counting function of prime triples as D*n/log(n)^3.
Decimal expansion of a constant related to the variance of the number of vertices of the largest tree associated with a random mapping on n symbols.
Decimal expansion of (1/2) Product_{p prime} 1+1/(p-1)^3, a constant related to I. M. Vinogradov’s proof of the ”ternary” Goldbach conjecture.
Decimal expansion of (6/Pi^2) Sum_{p prime} 1/(p(p+1)), a Meissel-Mertens constant related to the asymptotic density of certain sequences of integers.
Decimal expansion of C = log(2*Pi) + B_3 (where B_3 is A083343), one of Euler totient constants.
Decimal expansion of -1/(e^2 Ei(-1)), an increasing rooted tree enumeration constant associated with the Euler-Gompertz constant, where Ei is the exponential integral.
Decimal expansion of the variance of the degree (valency) of the root of a random rooted tree with n vertices.
Decimal expansion of p_3 (so named by S. Finch), a probability related to Vallée’s constant.
Decimal expansion of Product_{k >= 1} (k*(k+1))^(-1/(k*(k+1))), a constant related to the alternating Lüroth representations of real numbers.
Decimal expansion of lim_{N->infinity} (1/N^2 Sum_{n=1..N} K(n)), where K(n) is the squarefree kernel of n.
Decimal expansion of a function approximation constant which is the analog of Gibbs’s constant 2*G/Pi (A036793) for de la Vallée-Poussin sums.
First differences of 1/p(n), reciprocal of the number p(n) of unrestricted partitions of n (negated numerator).
First differences of 1/p(n), reciprocal of the number p(n) of unrestricted partitions of n (denominator).
Decimal expansion of ’kappa’, an asymptotic enumeration constant related to unit interval graphs.
Asymptotic mean (normalized by n) of the second longest cycle in a random permutation on n symbols.
Asymptotic variance (normalized by n^2) of the second longest cycle in a random permutation on n symbols.
Asymptotic mean (normalized by n) of the third longest cycle in a random permutation on n symbols.
Asymptotic variance (normalized by n^2) of the third longest cycle in a random permutation on n symbols.
Asymptotic mean (normalized by n) of the second largest connected component in a random mapping on n symbols.
Asymptotic variance (normalized by n^2) of the second largest connected component in a random mapping on n symbols.
Decimal expansion of s_4, a 4-dimensional Steiner ratio analog.
Decimal expansion of C_2 (so named by S. Finch), a constant which is an analog of Niven’s constant when mean of exponents is considered instead of maximum.
Decimal expansion of the first moment of the reciprocal gamma distribution.
Decimal expansion of the second moment of the reciprocal gamma distribution.
Decimal expansion of the probability that three positive integers are pairwise not coprime.
Decimal expansion of tau_2 (so named by S. Finch), the sum of squared eigenvalues of the Ruelle-Mayer linear operator G_2.
Decimal expansion of Rosser’s constant.
Decimal expansion of the radius of convergence of the generating function for the enumeration of rooted identity trees (A004111).
Decimal expansion of B, a constant appearing in an asymptotic formula related to the exponential divisor function sigma^(e).
Digital expansion of K_ccf, an analog of Khinchin’s constant for centered continued fractions.
Decimal expansion of the Klarner-Rivest polyomino constant.
Multiplicative with a(p^k) = p*a(k) for any prime p and k>0.
Decimal expansion of the mean number of iterations in a comparison algorithm using centered continued fractions, a constant related to Vallée’s constant.
Array of Markov triples (x,y,z) sorted by z, read by rows.
Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers.
Decimal expansion of Sum(1/p + 1/q) as (p, q) runs through the twin m^2 + 1 primes.
Given the two curves y = exp(-x) and y = 2/(exp(x) + exp(x/2)), draw a line tangent to both. This sequence is the decimal expansion of the (negated) x-coordinate of the point at which the line touches y = 2/(exp(x) + exp(x/2)).
Given the two curves y = exp(-x) and y = 2/(exp(x) + exp(x/2)), draw a line tangent to both. This sequence is the decimal expansion of the x-coordinate of the point at which the line touches y = exp(-x).
Unreduced numerator of expected rank of applicant in average rank secretary problem.
Given the two curves y = exp(-x) and y = 2/(exp(x) + exp(x/2)), draw a line tangent to both. This sequence is the decimal expansion of the y-coordinate of the point at which the line touches y = 2/(exp(x) + exp(x/2)).
Decimal expansion of the asymptotic value of the second raw moment of the maximal exponent in the prime factorizations of n (A051903).
Given the two curves y = (1 + exp(x))/2 and y = (1 + exp(x))/(1 + exp(x/2)), draw a line tangent to both. This sequence is the decimal expansion of the (negated) x-coordinate of the point at which the line touches y = (1 + exp(x))/2.
Given the two curves y = (1 + exp(x))/2 and y = (1 + exp(x))/(1 + exp(x/2)), draw a line tangent to both. This sequence is the decimal expansion of the x-coordinate of the point at which the line touches y = (1 + exp(x))/(1 + exp(x/2)).
Decimal expansion of Product_{p prime, p == 1 (mod 4)} (1 - 2/p^2).
Given the two curves y = (1 - exp(x/2))/(exp(x) + exp(x/2)) and y = (exp(-x) - 1)/2, draw a line tangent to both. This sequence is the decimal expansion of the y-coordinate (negated) of the point at which the line touches y = (1 - exp(x/2))/(exp(x) + exp(x/2)).
Given the two curves y = (1 - exp(x/2))/(exp(x) + exp(x/2)) and y = (exp(-x) - 1)/2, draw a line tangent to both. This sequence is the decimal expansion of the y-coordinate (negated) of the point at which the line touches y = (exp(-x) - 1)/2.
Decimal expansion of the Gaussian twin prime constant: the Hardy-Littlewood constant for A096012.
Decimal expansion of Shanks’s constant: the Hardy-Littlewood constant for A000068.
Decimal expansion of Lal’s constant: the Hardy-Littlewood constant for A217795.
Decimal expansion of Mertens constant C(5,1).
Decimal expansion of the Mertens constant C(5,4).
Decimal expansion of the angle in radians at the apex of the Calabi triangle.
Decimal expansion of the value of the Dickman function at phi + 1 = phi^2 = (3 + sqrt(5))/2 (A104457).
Decimal expansion of the value of the Buchstab function at phi + 2 = (5 + sqrt(5))/2 (A296184).
Decimal expansion of K, a constant arising in the analysis of the binary Euclidean algorithm.
The number of cubefull numbers (A036966) not exceeding 10^n.
Decimal expansion of Product_{p prime} (1 + 1/p^(4/3) + 1/p^(5/3)).
Decimal expansion of zeta(3/4) * Product_{p prime} (1 + 1/p^(5/4) - 1/p^2 - 1/p^(9/4)) (negated).
Decimal expansion of zeta(3/5) * zeta(4/5) * Product_{p prime} (1 - 1/p^(8/5) - 1/p^(9/5) - 1/p^2 + 1/p^(13/5) + 1/p^(14/5)).
Decimal expansion of the Mertens constant M(4,1) arising in the formula for the sum of reciprocals of primes p == 1 (mod 4) (negated).
Decimal expansion of the Mertens constant M(4,3) arising in the formula for the sum of reciprocals of primes p == 1 (mod 4).