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 A268895 Decimal expansion of the upper bound of 1/Gamma(x) - x on the unit interval x = [0,1]. 4
 7, 2, 1, 8, 6, 2, 7, 9, 6, 8, 2, 2, 4, 8, 1, 1, 6, 4, 9, 3, 4, 3, 7, 0, 1, 1, 4, 8, 8, 4, 6, 0, 0, 2, 8, 1, 1, 8, 7, 0, 1, 7, 7, 5, 4, 8, 9, 8, 1, 6, 1, 3, 9, 3, 8, 7, 4, 7, 3, 5, 8, 8, 3, 4, 8, 3, 9, 3, 8, 1, 4, 5, 8, 9, 1, 9, 3, 8, 6, 7, 2, 1, 5, 3, 3, 6, 3, 8, 9, 0, 2, 2, 0, 0, 8, 4, 8, 7, 6, 0, 7, 1, 0, 6 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET -1,1 COMMENTS Gamma(x) stands for the gamma function (Euler's integral of the second kind). On the unit interval the function 1/Gamma(x) may be bounded from below and from above as follows: x <= 1/Gamma(x) <= x + C, where C = 0.072186279... is the constant which we introduced above. Numerical simulations show that these lower and upper bounds are both quite accurate. Some other bounds for 1/Gamma(x) may be found in the reference given below. Numerically, the value of C is quite close to the first Stieltjes constant with the opposite sign (see A082633). LINKS Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi, arXiv:1408.3902 [math.NT], 2014-2016. FORMULA Equals 1/Gamma(x_0) - x_0, where x_0 is the unique positive root of the equation Gamma(x) + Psi(x) = 0 (see A268893). EXAMPLE 0.0721862796822481164934370114884600281187017754898161... MAPLE Digits:= 500; x0:=fsolve(Psi(x)+GAMMA(x)=0, x): evalf(1/GAMMA(x0)-x0, 120); MATHEMATICA y = FindRoot[Gamma[x]+PolyGamma[x]==0, {x, 0.6}, WorkingPrecision->120][[1, 2]]; N[1/Gamma[y] - y, 120] // RealDigits[#, 10, 104] & // First PROG (PARI) default(realprecision, 500); x0=solve(x = 0.60, 0.68, gamma(x)+psi(x)); 1/gamma(x0)-x0 CROSSREFS Cf. A268893, A268911. Sequence in context: A154020 A060991 A120455 * A108433 A274570 A334056 Adjacent sequences:  A268892 A268893 A268894 * A268896 A268897 A268898 KEYWORD nonn,cons AUTHOR Iaroslav V. Blagouchine, Feb 15 2016 STATUS approved

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Last modified July 31 06:15 EDT 2021. Contains 346369 sequences. (Running on oeis4.)