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

5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 79, 84, 89, 94, 99, 104, 109, 114, 119, 124, 129, 134, 139, 144, 149, 153, 158, 163, 168, 173, 178, 183, 188, 193, 198, 203, 208, 213, 218, 223, 227, 232, 237, 242, 247, 252, 257, 262, 267, 272, 277

Offset: 1

Author

Clark Kimberling, Oct 07 2014

Keywords

• nonn
• easy

Comments

It appears that a(n+1) - a(n) is in {4,5} for n >= 1.

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

Examples

			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:
n ... cos(Pi/a(n)) + 1/(n*a(n))
1 ... 1.009016994
2 ... 1.001056516
3 ... 1.000369823
4 ... 1.000188341
5 ... 1.000114701
6 ... 1.000077451
		

Links

• Clark Kimberling and Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (first 500 terms from Clark Kimberling)

Crossrefs

Cf. A102753, A248360.

Programs

Mathematica

z = 800; f[n_] := f[n] = Select[Range[z], Cos[Pi/#] + 1/(#*n) > 1 &, 1];
u = Flatten[Table[f[n], {n, 1, z}]]  (* A248359 *)
Table[Floor[1/(1 - Cos[Pi/n])], {n, 1, z/10}]  (* A248360 *)
Table[k=1; While[Cos[Pi/k]+1/(k*n)<=1,k++]; k,{n,1,100}] (* Vaclav Kotesovec, Oct 09 2014 *)

Formula

a(n) ~ n*Pi^2/2 = n*A102753. - Vaclav Kotesovec, Oct 09 2014