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A201573 Decimal expansion of greatest x satisfying x^2 + 6 = csc(x) and 0 < x < Pi.

%I A201573 #15 Feb 07 2025 16:44:07
%S A201573 3,0,7,6,8,9,4,9,2,9,2,4,6,1,9,2,0,2,3,1,6,6,6,9,3,6,4,7,3,2,7,7,2,5,
%T A201573 7,7,3,2,4,8,4,1,9,8,0,6,5,8,2,3,7,4,3,2,0,1,5,8,3,9,9,5,2,4,3,9,9,1,
%U A201573 1,1,5,7,6,0,6,3,1,5,1,1,6,6,3,2,3,5,4,5,1,8,1,1,9,1,2,3,5,6,5,9
%N A201573 Decimal expansion of greatest x satisfying x^2 + 6 = csc(x) and 0 < x < Pi.
%C A201573 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201573 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201573 least:  0.166669163175400949565200320627761299158167...
%e A201573 greatest:  3.076894929246192023166693647327725773248...
%t A201573 a = 1; c = 6;
%t A201573 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201573 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201573 r = x /. FindRoot[f[x] == g[x], {x, .1, .2}, WorkingPrecision -> 110]
%t A201573 RealDigits[r]   (* A201572 *)
%t A201573 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.1}, WorkingPrecision -> 110]
%t A201573 RealDigits[r]   (* A201573 *)
%o A201573 (PARI) a=1; c=6; solve(x=3, 3.1, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 21 2018
%Y A201573 Cf. A201564.
%K A201573 nonn,cons
%O A201573 1,1
%A A201573 _Clark Kimberling_, Dec 03 2011
%E A201573 Terms a(87) onward corrected by _G. C. Greubel_, Aug 21 2018