A248359 - OEIS

%I #21 Mar 04 2021 03:40:31

%S 5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,79,84,89,94,99,104,109,

%T 114,119,124,129,134,139,144,149,153,158,163,168,173,178,183,188,193,

%U 198,203,208,213,218,223,227,232,237,242,247,252,257,262,267,272,277

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

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

%C 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 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 a(n) ~ n*Pi^2/2 = n*A102753. - _Vaclav Kotesovec_, Oct 09 2014

%e 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 n ... cos(Pi/a(n)) + 1/(n*a(n))

%e 1 ... 1.009016994

%e 2 ... 1.001056516

%e 3 ... 1.000369823

%e 4 ... 1.000188341

%e 5 ... 1.000114701

%e 6 ... 1.000077451

%t z = 800; f[n_] := f[n] = Select[Range[z], Cos[Pi/#] + 1/(#*n) > 1 &, 1];

%t u = Flatten[Table[f[n], {n, 1, z}]]  (* A248359 *)

%t Table[Floor[1/(1 - Cos[Pi/n])], {n, 1, z/10}]  (* A248360 *)

%t Table[k=1; While[Cos[Pi/k]+1/(k*n)<=1,k++]; k,{n,1,100}] (* _Vaclav Kotesovec_, Oct 09 2014 *)

%Y Cf. A102753, A248360.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Oct 07 2014