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A202347 Decimal expansion of x < 0 satisfying x + e = exp(x).

%I A202347 #18 Feb 17 2025 02:31:26
%S A202347 2,6,4,7,4,5,0,2,4,2,0,4,9,9,6,6,7,2,0,7,2,7,2,0,1,2,2,2,1,4,6,4,1,5,
%T A202347 2,4,3,5,5,9,2,9,7,3,7,7,0,8,0,1,9,6,6,8,3,0,5,4,0,3,2,2,2,7,8,8,5,8,
%U A202347 1,1,9,4,6,0,7,5,9,2,2,7,8,4,5,5,2,1,4,9,0,3,3,5,7,2,7,8,8,0,3
%N A202347 Decimal expansion of x < 0 satisfying x + e = exp(x).
%C A202347 See A202320 for a guide to related sequences. The Mathematica program includes a graph.
%H A202347 G. C. Greubel, <a href="/A202347/b202347.txt">Table of n, a(n) for n = 1..10000</a>
%H A202347 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A202347 Equals -exp(1) - LambertW(-exp(-exp(1))). - _G. C. Greubel_, Nov 09 2017
%e A202347 x < 0: -2.64745024204996672072720122214641524...
%e A202347 x > 0: 1.420370118020083458458421283899772980...
%t A202347 u = 1; v = E;
%t A202347 f[x_] := u*x + v; g[x_] := E^x
%t A202347 Plot[{f[x], g[x]}, {x, -3, 2}, {AxesOrigin -> {0, 0}}]
%t A202347 r = x /. FindRoot[f[x] == g[x], {x, -2.7, -2.6}, WorkingPrecision -> 110]
%t A202347 RealDigits[r] (* A202347 *)
%t A202347 r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
%t A202347 RealDigits[r] (* A104689 *)
%t A202347 RealDigits[-E - LambertW[-Exp[-E]], 10, 100][[1]] (* _G. C. Greubel_, Nov 09 2017 *)
%o A202347 (PARI) solve(x=-3, 0, exp(x)-exp(1)-x) \\ _Michel Marcus_, Nov 09 2017
%Y A202347 Cf. A202320.
%K A202347 nonn,cons
%O A202347 1,1
%A A202347 _Clark Kimberling_, Dec 17 2011