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A202351 Decimal expansion of least x satisfying 3*x = exp(x).

%I A202351 #23 Nov 21 2024 15:36:02
%S A202351 6,1,9,0,6,1,2,8,6,7,3,5,9,4,5,1,1,2,1,5,2,3,2,6,9,9,4,0,2,0,9,2,2,2,
%T A202351 3,3,3,0,1,4,7,1,7,7,7,2,6,2,9,6,9,3,5,2,4,5,9,8,3,6,0,7,4,4,9,2,9,3,
%U A202351 7,3,5,2,2,5,5,0,8,8,7,3,4,6,1,1,0,4,6,9,2,6,1,8,8,2,5,8,8,4,0
%N A202351 Decimal expansion of least x satisfying 3*x = exp(x).
%C A202351 See A202320 for a guide to related sequences. The Mathematica program includes a graph.
%H A202351 G. C. Greubel, <a href="/A202351/b202351.txt">Table of n, a(n) for n = 0..9999</a> [Offset shifted by _Georg Fischer_, Oct 18 2021]
%H A202351 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%e A202351 least:  0.61906128673594511215232699402092223330147...
%e A202351 greatest:  1.51213455165784247389673967807203870460...
%t A202351 u = 3; v = 0;
%t A202351 f[x_] := u*x + v; g[x_] := E^x
%t A202351 Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
%t A202351 r = x /. FindRoot[f[x] == g[x], {x, 0.6, 0.7}, WorkingPrecision -> 110]
%t A202351 RealDigits[r] (* A202351 *)
%t A202351 r = x /. FindRoot[f[x] == g[x], {x, 1.5, 1.6}, WorkingPrecision -> 110]
%t A202351 RealDigits[r] (* A202352 *)
%t A202351 RealDigits[ -ProductLog[-1/3], 10, 99] // First (* _Jean-François Alcover_, Feb 27 2013 *)
%o A202351 (PARI) solve(x=0, 1, 3*x-exp(x)) \\ _Michel Marcus_, Nov 09 2017
%Y A202351 Cf. A202320.
%K A202351 nonn,cons
%O A202351 0,1
%A A202351 _Clark Kimberling_, Dec 17 2011
%E A202351 Offset corrected by _Georg Fischer_, Aug 02 2021