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A248106
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Least k such that ((k+1)/(k-1))^k - e^2 < 1/n^2.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 14.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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