A346033 -id:A346033 - cp's OEIS frontend

A345670 a(n) is the smallest integer k > 0 such that 10^(-n-1) < |cos(k) - round(cos(k))| < 10^(-n).

1, 3, 11, 44, 22, 16685, 5325, 1775, 710, 355, 104348, 312689, 1146408, 20530996, 10838702, 5419351, 165707065, 411557987

Offset: 0

Treanungkur Mal, Jul 02 2021

			For n = 4, a(n) = 22 because 22 is the smallest positive integer k such that 10^(-5) < |cos(k) - round(cos(k))| < 10^(-4): |cos(22) - round(cos(22))| = 0.0000391...

Cf. A346033 (sin), A345404 (tan).

a(n) = my(k=1, ok=0, x); while (!ok, x=abs(cos(k) - round(cos(k))); ok = (x>1/10^(n+1)) && (x < 1/10^n); if (ok, break); k++); k; \\ Michel Marcus, Jul 02 2021

a(16)-a(17) from Jon E. Schoenfield and Sean A. Irvine, Jul 03 2021

A346151 a(n) is the smallest integer k > 0 such that 1 - tanh(k) < 10^(-n).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 75, 76, 77

Offset: 0

Treanungkur Mal, Jul 07 2021

nonn

As k increases, 1 - tanh(k) rapidly approaches 2*exp(-2*k), and the smallest integer k such that 2*exp(-2*k) < 10^(-n), i.e., such that k > (n*log(10) + log(2))/2, is simply ceiling((1/2)*(n*log(10) + log(2))). It seems very likely that this expression gives a(n) for all n >= 0. - Jon E. Schoenfield, Jul 08 2021

			For n = 3, a(3) = 4 because 4 is the smallest positive integer k such that 1 - tanh(k) < 10^(-3): 1 - tanh(4) = 0.00067....

Cf. A346033 (sin), A345670 (cos).

a[0] = 1; a[n_] := Ceiling @ ArcTanh[1 - 10^(-n)]; Array[a, 100, 0] (* Amiram Eldar, Jul 12 2021 *)

cp's OEIS Frontend

A345670 a(n) is the smallest integer k > 0 such that 10^(-n-1) < |cos(k) - round(cos(k))| < 10^(-n).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Extensions