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A248515 Least number k such that 1 - k*sin(1/k) < 1/n^2. 3
1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
This sequences provides insight into the manner of convergence of n*sin(1/n). One may also consider: [1/(1 - n*sin(1/n)] = 6*n^2 = A033581(n) for n >= 1.
a(n+1) - a(n) is in {0,1} for n >= 1, so that the position sequences A138235 and A022840 partition the positive integers.
a(n) = A194986(n). - Clark Kimberling, Jan 15 2015
LINKS
FORMULA
a(n) = ceiling (n/sqrt(6)) for n >= 1.
EXAMPLE
Approximations:
n 1-k*sin(1/k) 1/n^2
1 0.158529 1
2 0.041148 0.25
3 0.018415 0.11111
4 0.010384 0.0625
5 0.006653 0.04
a(5) = 3 because 1 - 3*sin(1/3) < 1/25 < 1 - 2*sin(1/2).
MATHEMATICA
z = 120; p[k_] := p[k] = k*Sin[1/k]; N[Table[1 - p[n], {n, 1, z/5}]]
f[n_] := f[n] = Select[Range[z], 1 - p[#] < 1/n^2 &, 1];
u = Flatten[Table[f[n], {n, 1, z}]] (* A248515 *)
v = Flatten[Position[Differences[u], 0]] (* A138235 *)
w = Flatten[Position[Differences[u], 1]] (* A022840 *)
Table[Ceiling[n / Sqrt[6]], {n, 70}] (* Vincenzo Librandi, Jun 17 2015 *)
PROG
(Magma) [Ceiling(n/Sqrt(6)): n in [1..70]]; // Vincenzo Librandi, Jun 17 2015
CROSSREFS
Sequence in context: A085269 A173276 A288156 * A194986 A302128 A054071
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Oct 08 2014
STATUS
approved

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Last modified June 25 18:50 EDT 2024. Contains 373707 sequences. (Running on oeis4.)