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A248359 Least number k such that cos(Pi/k) + 1/(k*n) > 1.

%I A248359 #21 Mar 04 2021 03:40:31
%S A248359 5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,79,84,89,94,99,104,109,
%T A248359 114,119,124,129,134,139,144,149,153,158,163,168,173,178,183,188,193,
%U A248359 198,203,208,213,218,223,227,232,237,242,247,252,257,262,267,272,277
%N A248359 Least number k such that cos(Pi/k) + 1/(k*n) > 1.
%C A248359 It appears that a(n+1) - a(n) is in {4,5} for n >= 1.
%C A248359 Lim_{n->infinity} a(n)/n = Pi^2/2 = 4.9348022..., but lim_{n->infinity} (a(n+1) - a(n)) does not exist; Pi^2/2 is only a mean value of these differences. - _Vaclav Kotesovec_, Oct 09 2014
%H A248359 Clark Kimberling and Vaclav Kotesovec, <a href="/A248359/b248359.txt">Table of n, a(n) for n = 1..10000</a> (first 500 terms from Clark Kimberling)
%F A248359 a(n) ~ n*Pi^2/2 = n*A102753. - _Vaclav Kotesovec_, Oct 09 2014
%e A248359 Taking n = 2, we have cos(Pi/9) + 1/(18) = 0.99524... < 1 < 1.0010565... = cos(Pi/10) + 1/(20), so that a(2) = 10, as corroborated for n = 2 in the following list of approximations:
%e A248359 n ... cos(Pi/a(n)) + 1/(n*a(n))
%e A248359 1 ... 1.009016994
%e A248359 2 ... 1.001056516
%e A248359 3 ... 1.000369823
%e A248359 4 ... 1.000188341
%e A248359 5 ... 1.000114701
%e A248359 6 ... 1.000077451
%t A248359 z = 800; f[n_] := f[n] = Select[Range[z], Cos[Pi/#] + 1/(#*n) > 1 &, 1];
%t A248359 u = Flatten[Table[f[n], {n, 1, z}]]  (* A248359 *)
%t A248359 Table[Floor[1/(1 - Cos[Pi/n])], {n, 1, z/10}]  (* A248360 *)
%t A248359 Table[k=1; While[Cos[Pi/k]+1/(k*n)<=1,k++]; k,{n,1,100}] (* _Vaclav Kotesovec_, Oct 09 2014 *)
%Y A248359 Cf. A102753, A248360.
%K A248359 nonn,easy
%O A248359 1,1
%A A248359 _Clark Kimberling_, Oct 07 2014