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 A257396 Decimal expansion of the negated x-coordinate of the inflection point of product{1 + x^k, k >= 1} that has maximal x-coordinate. 4
 2, 3, 2, 3, 3, 7, 7, 3, 1, 6, 5, 8, 6, 7, 3, 0, 8, 0, 4, 8, 7, 9, 7, 2, 5, 4, 7, 7, 1, 7, 5, 3, 5, 3, 9, 8, 1, 4, 2, 4, 0, 4, 4, 7, 6, 9, 2, 3, 4, 5, 5, 9, 8, 9, 3, 4, 1, 8, 8, 7, 9, 0, 1, 1, 8, 0, 8, 4, 0, 3, 1, 6, 3, 4, 2, 3, 5, 8, 2, 8, 5, 5, 3, 9, 0, 6, 0, 6, 7, 5, 6, 1, 9, 0, 2, 8, 2, 5, 7, 6, 2, 0, 3, 4 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS The function product{1 + x^k, k >= 1} has two inflection points:  (-0.78983..., 0.17671...) and (-0.23233..., 0.80084...). LINKS EXAMPLE x = -0.2323377316586730804879725477175353981424044769... MATHEMATICA f[x_] := f[x] = Product[(1 + x^k), {k, 1, 1000}]; p[x_, z_] := Sum[n/(x + x^(1 - n)), {n, z}]^2 + Sum[(n*x^(n - 2)*(n - x^n - 1))/(1 + x^n)^2, {n, z}]; Plot[f[x], {x, -1, 1}] (* plot showing 2 infl. pts. *) t = x /. FindRoot[p[x, 1000], {x, -0.8}, WorkingPrecision -> 100] (* A257394 *) u = f[t] (* A257395 *) v = x /. FindRoot[p[x, 200], {x, -0.3}, WorkingPrecision -> 100]  (* A257396 *) w = f[v] (* A257397 *) RealDigits[t, 10][[1]]  (* A257394 *) RealDigits[u, 10][[1]]  (* A257395 *) RealDigits[v, 10][[1]]  (* A257396 *) RealDigits[w, 10][[1]]  (* A257397 *) (* Peter J. C. Moses, Apr 21 2015 *) digits = 104; QP = QPochhammer; QPP[x_] := With[{dx = 10^-digits}, (QP[-1, x+dx] - QP[-1, x-dx])/(4*dx)]; x0 = x /. NMinimize[{QPP[x], -1/2 < x < 0}, x, WorkingPrecision -> 4 digits][[2]]; RealDigits[x0, 10, digits] // First (* Jean-François Alcover, Nov 19 2015 *) CROSSREFS Cf. A257394, A257395, A257397. Sequence in context: A003051 A305866 A328406 * A293519 A237582 A097352 Adjacent sequences:  A257393 A257394 A257395 * A257397 A257398 A257399 KEYWORD nonn,cons,easy AUTHOR Clark Kimberling, Apr 22 2015 EXTENSIONS More digits from Jean-François Alcover, Nov 19 2015 STATUS approved

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Last modified December 5 13:26 EST 2019. Contains 329751 sequences. (Running on oeis4.)