A248418 Least number k such that k*tan(Pi/k) - Pi < 1/n.

6, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 26, 26, 26, 26, 26, 27, 27, 27

Offset

3,1

Comments

a(n+1) - a(n) is in {0,1} for n >= 1, so that the position sequences A248420 and A248421 partition the positive integers.

Links

Clark Kimberling, Table of n, a(n) for n = 3..2000

Formula

a(n) ~ Pi * sqrt(Pi*n/3). - Vaclav Kotesovec, Oct 09 2014

Example

Approximations:
n ... k*tan(Pi/k)-Pi ... 1/n
3 ... 2.05456 .......... 0.33333
4 ... 0.85840 .......... 0.25
5 ... 0.49112 .......... 0.2
6 ... 0.32250 .......... 0.16666
7 ... 0.22943 .......... 0.14285
a(4) = 7 because 7*tan(Pi/7) < 1/4 < 6*tan(Pi/6).

Mathematica

z = 550; p[k_] := p[k] = k*Tan[Pi/k]; N[Table[p[n] - Pi, {n, 3, z/10}]]
f[n_] := f[n] = Select[2 + Range[z], p[#] - Pi < 1/n &, 1];
u = Flatten[Table[f[n], {n, 3, z}]]  (* A248418 *)
Differences[u]
Flatten[Position[Differences[u], 0]]     (* A248419 *)
Flatten[Position[Differences[u], 1]]     (* A248420 *)
Table[Floor[1/(p[n] - Pi)], {n, 3, z}]   (* A248421 *)

Crossrefs

Cf. A248419, A248420, A248421.

Sequence in context: A335094 A153604 A324031 * A168679 A102304 A114459

Adjacent sequences: A248415 A248416 A248417 * A248419 A248420 A248421

Keyword

nonn,easy

Author

Clark Kimberling, Oct 07 2014

Status

approved