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A248111 Least k such that ((k+2)/(k-2))^k - e^4 < 1/n.

%I A248111 #4 Oct 02 2014 22:36:21
%S A248111 18,25,30,35,39,42,46,49,52,55,57,60,62,64,67,69,71,73,75,77,79,81,82,
%T A248111 84,86,88,89,91,92,94,96,97,99,100,101,103,104,106,107,108,110,111,
%U A248111 112,114,115,116,118,119,120,121,122,124,125,126,127,128,129,130
%N A248111 Least k such that ((k+2)/(k-2))^k - e^4 < 1/n.
%C A248111 In general, for fixed positive m, the limit of ((m*x+1)/(m*x-1))^x is e^(2/m), as illustrated by A248103, A248106, A248111.
%D A248111 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 14.
%H A248111 Clark Kimberling, <a href="/A248111/b248111.txt">Table of n, a(n) for n = 1..1000</a>
%e A248111 Approximations are shown here:
%e A248111 n ... ((n+2)/(n-2))^n - e^4 ... 1/n
%e A248111 6 .... 9.40185 ................ 0.16666
%e A248111 12 ... 2.09576 ................ 0.8333333
%e A248111 18 ... 0.913001 ............... 0.0555555
%e A248111 24 ... 0.510023 ............... 0.0416667
%e A248111 30 ... 0.325376 ............... 0.0333333
%e A248111 36 ... 0.225565 ............... 0.0277778
%e A248111 a(2) = 25 because p(25) - e^4 < 1/2 < p(24) - e^4.
%t A248111 z = 1200; p[k_] := p[k] = ((k + 2)/(k - 2))^k
%t A248111 N[Table[p[n] - E^4, {n, 1, z/12}]]
%t A248111 f[n_] := f[n] = Select[Range[z], # > 2 && p[#] - E^4 < 1/n &, 1]
%t A248111 u = Flatten[Table[f[n], {n, 1, z/10}]]  (* A248111 *)
%Y A248111 Cf. A248103, A248111.
%K A248111 nonn,easy
%O A248111 1,1
%A A248111 _Clark Kimberling_, Oct 02 2014