A247985 Least number k such that product{(k^2 + h)/(k^2 - h), h = 1..k} - e < 1/n.

4, 7, 9, 12, 15, 17, 20, 23, 26, 28, 31, 34, 37, 39, 42, 45, 47, 50, 53, 56, 58, 61, 64, 66, 69, 72, 75, 77, 80, 83, 85, 88, 91, 94, 96, 99, 102, 104, 107, 110, 113, 115, 118, 121, 123, 126, 129, 132, 134, 137, 140, 143, 145, 148, 151, 153, 156, 159, 162

Offset: 1

Author

Clark Kimberling, Sep 29 2014

Keywords

• nonn
• easy

Comments

a(n+1) - a(n) is in {2,3} for n >= 1.

Examples

			Let w(n) = product{(n^2 + h)/(n^2 - h), h = 1..n} - e
Approximations are shown here:
n ... w(n)  ........ 1/n
1 ... undefined .... 1
2 ... 2.28172 .....  0.5
3 ... 1.21029 ...... 0.333333
4 ... 0.831169 ..... 0.25
5 ... 0.634485 ..... 0.2
6 ... 0.513554 ..... 0.166666
7 ... 0.431526 ..... 0.142857
a(2) = 7 because w(7) < 1/2 < w(6).
		

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. A247986, A247987.

Programs

Mathematica

z = 100; p[k_] := p[k] = Product[(k^2 + h)/(k^2 - h), {h, 1, k}] (* Finch p. 14 *)
N[Table[p[n] - E, {n, 2, z}]]
f[n_] := f[n] = Select[1 + Range[z], p[#] - E < 1/n &, 1];
u = Flatten[Table[f[n], {n, 1, z}]] ;  (* A247985 *)
v = Differences[u];
Flatten[Position[v, 2]]; (* A247986 *)
Flatten[Position[v, 3]]; (* A247987 *)