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A201589 Decimal expansion of least x satisfying 5*x^2 = csc(x) and 0 < x < Pi.

%I A201589 #13 Feb 07 2025 16:44:07
%S A201589 5,9,6,6,2,6,8,1,9,8,6,0,7,0,4,5,4,6,7,6,1,8,3,2,8,5,9,0,8,2,1,4,1,0,
%T A201589 4,8,3,0,3,6,5,3,1,0,0,8,7,0,2,9,3,0,5,7,4,4,7,1,8,2,0,4,7,7,5,8,3,7,
%U A201589 4,7,8,6,0,6,4,1,9,9,1,6,3,4,1,9,4,0,7,6,9,5,4,7,5,8,8,9,5,2,2,7,8
%N A201589 Decimal expansion of least x satisfying 5*x^2 = csc(x) and 0 < x < Pi.
%C A201589 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201589 G. C. Greubel, <a href="/A201589/b201589.txt">Table of n, a(n) for n = 0..10000</a>
%H A201589 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201589 least:  0.596626819860704546761832859082141048303653100...
%e A201589 greatest:  3.121059463523827415360175700034092048910749...
%t A201589 a = 5; c = 0;
%t A201589 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201589 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201589 r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
%t A201589 RealDigits[r]   (* A201589 *)
%t A201589 r = x /. FindRoot[f[x] == g[x], {x, 3.1, 3.14}, WorkingPrecision -> 110]
%t A201589 RealDigits[r]   (* A201590 *)
%o A201589 (PARI) a=5; c=0; solve(x=0.5, 1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 22 2018
%Y A201589 Cf. A201564.
%K A201589 nonn,cons
%O A201589 0,1
%A A201589 _Clark Kimberling_, Dec 03 2011
%E A201589 Terms a(90) onward corrected by _G. C. Greubel_, Aug 22 2018