Treanungkur Mal - cp's OEIS frontend

A346166 a(n) is the smallest integer k >= 0 such that cosh(k) > 10^(n^2).

1, 3, 10, 22, 38, 59, 84, 114, 149, 188, 231, 280, 333, 390, 452, 519, 591, 667, 747, 832, 922, 1017, 1116, 1219, 1327, 1440, 1558, 1680, 1806, 1938, 2074, 2214, 2359, 2509, 2663, 2822, 2985, 3153, 3326, 3503, 3685, 3872, 4063, 4259, 4459, 4664, 4873, 5088

Offset: 0

Treanungkur Mal, Jul 08 2021

			For n = 4, a(4) = 38 because 38 is the smallest positive integer k such that cosh(k) > 10^(16): cosh(38) = 1.59...*10^16, which is greater than 10^(4^2), i.e., 10^16.

a[0] = 1; a[n_] := Ceiling @ ArcCosh[10^(n^2)]; Array[a, 50, 0] (* Amiram Eldar, Jul 12 2021 *)

a(n) = 1 + floor(acosh(10^(n^2))); \\ Michel Marcus, Aug 24 2021

a(n) = 1 + floor(arccosh(10^(n^2))). - Jon E. Schoenfield, Jul 08 2021

More terms from Jinyuan Wang, Jul 08 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 *)

A346169 a(n) = floor(cosh(n+1) - cosh(n)).

Original entry on oeis.org

0, 2, 6, 17, 46, 127, 346, 942, 2561, 6961, 18923, 51440, 139829, 380095, 1033206, 2808546, 7634421, 20752508, 56411165, 153341447, 416825269, 1133048555, 3079945300, 8372159341, 22757888603, 61862355045, 168159315586, 457104411844, 1242538616426

Offset: 0

Treanungkur Mal, Jul 08 2021

nonn

			For n = 3, a(3) = 17 because the difference between cosh(4) and cosh(3) is 17.24057..., and floor(17.24057...) = 17.

Cf. A000501.

a[n_] := Floor[Cosh[n+1] - Cosh[n]]; Array[a, 30,0] (* Amiram Eldar, Jul 12 2021 *)

More terms from Jinyuan Wang, Jul 08 2021

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

Original entry on oeis.org

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

A346033 a(n) is the smallest integer k > 0 such that 10^(-n-1) < |sin(k) - round(sin(k))| < 10^(-n).

Original entry on oeis.org

1, 2, 14, 55, 33, 11, 35489, 46849, 50754, 51819, 52174, 260515, 1719612, 573204, 21104200, 37362253, 42781604, 122925461, 534483448

Offset: 0

Treanungkur Mal, Jul 01 2021

			For n = 3, a(n) = 55 because 55 is the smallest positive integer k such that 10^(-4) < |sin(k) - round(sin(k))| < 10^(-3): |sin(55) - round(sin(55))| = 0.000244....

Cf. A096456, A345404.

a(16)-a(18) from Jon E. Schoenfield and Sean A. Irvine, Jul 02 2021

cp's OEIS Frontend

User: Treanungkur Mal

A346166 a(n) is the smallest integer k >= 0 such that cosh(k) > 10^(n^2).

Views

Author

Keywords

Examples

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A346169 a(n) = floor(cosh(n+1) - cosh(n)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions

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

A346033 a(n) is the smallest integer k > 0 such that 10^(-n-1) < |sin(k) - round(sin(k))| < 10^(-n).

Views

Author

Keywords

Examples

Crossrefs

Extensions