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A201572 Decimal expansion of least x satisfying x^2 + 6 = csc(x) and 0 < x < Pi.

%I A201572 #13 Feb 07 2025 16:44:07
%S A201572 1,6,6,6,6,9,1,6,3,1,7,5,4,0,0,9,4,9,5,6,5,2,0,0,3,2,0,6,2,7,7,6,1,2,
%T A201572 9,9,1,5,8,1,6,7,6,7,4,4,7,5,4,0,3,4,4,9,3,4,8,0,4,8,4,6,7,5,6,4,4,2,
%U A201572 3,4,9,3,7,1,3,9,6,9,9,0,8,9,8,0,2,6,1,8,0,1,5,2,8,3,7,0,8,7,0,8,0
%N A201572 Decimal expansion of least x satisfying x^2 + 6 = csc(x) and 0 < x < Pi.
%C A201572 See A201564 for a guide to related sequences. The Mathematica program includes a graph.
%H A201572 G. C. Greubel, <a href="/A201572/b201572.txt">Table of n, a(n) for n = 0..10000</a>
%H A201572 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201572 least:  0.166669163175400949565200320627761299158167...
%e A201572 greatest:  3.076894929246192023166693647327725773248...
%t A201572 a = 1; c = 6;
%t A201572 f[x_] := a*x^2 + c; g[x_] := Csc[x]
%t A201572 Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
%t A201572 r = x /. FindRoot[f[x] == g[x], {x, .1, .2}, WorkingPrecision -> 110]
%t A201572 RealDigits[r]   (* A201572 *)
%t A201572 r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.1}, WorkingPrecision -> 110]
%t A201572 RealDigits[r]   (* A201573 *)
%o A201572 (PARI) a=1; c=6; solve(x=0.1, .2, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 21 2018
%Y A201572 Cf. A201564.
%K A201572 nonn,cons
%O A201572 0,2
%A A201572 _Clark Kimberling_, Dec 03 2011
%E A201572 Terms a(83) onward corrected by _G. C. Greubel_, Aug 21 2018