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

18, 25, 30, 35, 39, 42, 46, 49, 52, 55, 57, 60, 62, 64, 67, 69, 71, 73, 75, 77, 79, 81, 82, 84, 86, 88, 89, 91, 92, 94, 96, 97, 99, 100, 101, 103, 104, 106, 107, 108, 110, 111, 112, 114, 115, 116, 118, 119, 120, 121, 122, 124, 125, 126, 127, 128, 129, 130

Offset: 1

Author

Clark Kimberling, Oct 02 2014

Keywords

• nonn
• easy

Comments

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.

Examples

			Approximations are shown here:
n ... ((n+2)/(n-2))^n - e^4 ... 1/n
6 .... 9.40185 ................ 0.16666
12 ... 2.09576 ................ 0.8333333
18 ... 0.913001 ............... 0.0555555
24 ... 0.510023 ............... 0.0416667
30 ... 0.325376 ............... 0.0333333
36 ... 0.225565 ............... 0.0277778
a(2) = 25 because p(25) - e^4 < 1/2 < p(24) - e^4.
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 14.

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A248103, A248111.

Programs

Mathematica

z = 1200; p[k_] := p[k] = ((k + 2)/(k - 2))^k
N[Table[p[n] - E^4, {n, 1, z/12}]]
f[n_] := f[n] = Select[Range[z], # > 2 && p[#] - E^4 < 1/n &, 1]
u = Flatten[Table[f[n], {n, 1, z/10}]]  (* A248111 *)