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A196617 Decimal expansion of the least x>0 satisfying 1 = (x^2)*sin(x).

%I A196617 #14 Jan 30 2025 13:55:37
%S A196617 1,0,6,8,2,2,3,5,4,4,1,9,7,2,4,9,0,1,8,2,8,3,4,7,1,1,1,4,2,6,3,0,9,2,
%T A196617 8,9,8,4,6,8,9,3,5,1,3,0,5,1,5,1,1,6,6,3,4,3,9,3,2,7,1,1,7,8,1,1,1,1,
%U A196617 7,7,2,9,7,6,4,7,3,2,9,6,6,3,4,9,8,5,4,8,2,3,1,4,9,6,1,9,0,7,1,0
%N A196617 Decimal expansion of the least x>0 satisfying 1 = (x^2)*sin(x).
%C A196617 This number is the least x>0 for which there exists a constant c such that the graph of y=cos(x) is tangent to the graph of the hyperbola y=(1/x)-c, as indicated by the graph in the Mathematica program.
%H A196617 G. C. Greubel, <a href="/A196617/b196617.txt">Table of n, a(n) for n = 1..10000</a>
%H A196617 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A196617 1.0682235441972490182834711142630928984689...
%t A196617 Plot[{1/x - .4544, Cos[x]}, {x, 0, 2 Pi}]
%t A196617 xt = x /. FindRoot[x^(-2) == Sin[x], {x, .5, .8}, WorkingPrecision -> 100]
%t A196617 RealDigits[xt]      (* A196617 *)
%t A196617 Cos[xt]
%t A196617 RealDigits[Cos[xt]] (* A196618 *)
%t A196617 c = N[1/xt - Cos[xt], 100]
%t A196617 RealDigits[c]       (* A196619 *)
%t A196617 slope = -Sin[xt]
%t A196617 RealDigits[slope]   (* A196620 *)
%o A196617 (PARI) a=1; c=0; solve(x=1, 1.5, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 22 2018
%Y A196617 Cf. A196619, A196612.
%K A196617 nonn,cons
%O A196617 1,3
%A A196617 _Clark Kimberling_, Oct 05 2011
%E A196617 Terms a(88) onward corrected by _G. C. Greubel_, Aug 22 2018