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A202352 Decimal expansion of greatest x satisfying 3*x = exp(x).

%I A202352 #19 Nov 21 2024 15:36:12
%S A202352 1,5,1,2,1,3,4,5,5,1,6,5,7,8,4,2,4,7,3,8,9,6,7,3,9,6,7,8,0,7,2,0,3,8,
%T A202352 7,0,4,6,0,3,6,5,0,3,8,5,1,3,5,3,5,9,4,5,4,2,5,9,2,8,5,4,7,3,9,9,8,9,
%U A202352 7,7,1,6,0,5,1,1,5,7,4,8,2,7,3,2,4,2,6,5,4,8,8,1,5,2,7,7,9,8,3
%N A202352 Decimal expansion of greatest x satisfying 3*x = exp(x).
%C A202352 See A202320 for a guide to related sequences. The Mathematica program includes a graph.
%H A202352 G. C. Greubel, <a href="/A202352/b202352.txt">Table of n, a(n) for n = 1..10000</a>
%H A202352 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A202352 Equals -LambertW(-1,-1/3). - _Gleb Koloskov_, Jun 12 2021
%e A202352 least:  0.61906128673594511215232699402092223330147...
%e A202352 greatest:  1.51213455165784247389673967807203870460...
%t A202352 u = 3; v = 0;
%t A202352 f[x_] := u*x + v; g[x_] := E^x
%t A202352 Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
%t A202352 r = x /. FindRoot[f[x] == g[x], {x, 0.6, 0.7}, WorkingPrecision -> 110]
%t A202352 RealDigits[r] (* A202351 *)
%t A202352 r = x /. FindRoot[f[x] == g[x], {x, 1.5, 1.6}, WorkingPrecision -> 110]
%t A202352 RealDigits[r] (* A202352 *)
%t A202352 RealDigits[ -ProductLog[-1, -1/3], 10, 99] // First (* _Jean-François Alcover_, Feb 27 2013 *)
%o A202352 (PARI) solve(x=1, 2, 3*x-exp(x)) \\ _Michel Marcus_, Nov 09 2017
%Y A202352 Cf. A202320.
%K A202352 nonn,cons
%O A202352 1,2
%A A202352 _Clark Kimberling_, Dec 17 2011