A248106 Least k such that ((k+1)/(k-1))^k - e^2 < 1/n^2.

3, 5, 7, 9, 12, 14, 16, 18, 20, 23, 25, 27, 29, 32, 34, 36, 38, 40, 43, 45, 47, 49, 52, 54, 56, 58, 60, 63, 65, 67, 69, 72, 74, 76, 78, 80, 83, 85, 87, 89, 92, 94, 96, 98, 100, 103, 105, 107, 109, 111, 114, 116, 118, 120, 123, 125, 127, 129, 131, 134, 136

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+1)/(n-1))^n - e^2 ... 1/n^2
2 ... 1.610943901 ............. 0.25
3 ... 0.610943901 ............. 0.11111
4 ... 0.326993283 ............. 0.0625
5 ... 0.204693901 ............. 0.04
6 ... 0.140479901 ............. 0.02777
a(2) = 5 because p(5) - e^2 < 1/4 < p(4) - e^2.
		

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 + 1)/(k - 1))^k; (* Finch p. 15 *);
N[Table[p[n] - E^2, {n, 2, z/20}]]
f[n_] := f[n] = Select[Range[z], # > 1 && p[#] - E^2 < 1/n^2 &, 1]
u = Flatten[Table[f[n], {n, 1, z/4}]]  (* A248106 *)