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

%I A247985 #5 Sep 30 2014 15:56:55
%S A247985 4,7,9,12,15,17,20,23,26,28,31,34,37,39,42,45,47,50,53,56,58,61,64,66,
%T A247985 69,72,75,77,80,83,85,88,91,94,96,99,102,104,107,110,113,115,118,121,
%U A247985 123,126,129,132,134,137,140,143,145,148,151,153,156,159,162
%N A247985 Least number k such that product{(k^2 + h)/(k^2 - h), h = 1..k} - e < 1/n.
%C A247985 a(n+1) - a(n) is in {2,3} for n >= 1.
%D A247985 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 14.
%H A247985 Clark Kimberling, <a href="/A247985/b247985.txt">Table of n, a(n) for n = 1..1000</a>
%e A247985 Let w(n) = product{(n^2 + h)/(n^2 - h), h = 1..n} - e
%e A247985 Approximations are shown here:
%e A247985 n ... w(n)  ........ 1/n
%e A247985 1 ... undefined .... 1
%e A247985 2 ... 2.28172 .....  0.5
%e A247985 3 ... 1.21029 ...... 0.333333
%e A247985 4 ... 0.831169 ..... 0.25
%e A247985 5 ... 0.634485 ..... 0.2
%e A247985 6 ... 0.513554 ..... 0.166666
%e A247985 7 ... 0.431526 ..... 0.142857
%e A247985 a(2) = 7 because w(7) < 1/2 < w(6).
%t A247985 z = 100; p[k_] := p[k] = Product[(k^2 + h)/(k^2 - h), {h, 1, k}] (* Finch p. 14 *)
%t A247985 N[Table[p[n] - E, {n, 2, z}]]
%t A247985 f[n_] := f[n] = Select[1 + Range[z], p[#] - E < 1/n &, 1];
%t A247985 u = Flatten[Table[f[n], {n, 1, z}]] ;  (* A247985 *)
%t A247985 v = Differences[u];
%t A247985 Flatten[Position[v, 2]]; (* A247986 *)
%t A247985 Flatten[Position[v, 3]]; (* A247987 *)
%Y A247985 Cf. A247986, A247987.
%K A247985 nonn,easy
%O A247985 1,1
%A A247985 _Clark Kimberling_, Sep 29 2014