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

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

0,2

Comments

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

Links

Table of n, a(n) for n=0..66.

Example

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....

Mathematica

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

Prog

(C++)
/* Only suitable for computing a(0) to a(14) due to double precision limits. */
#include <iostream>
#include <cmath>
using namespace std;
int main(int argc, char** argv) {
    int control = 1;
    for (int n=0; n<=14; n++) {
        for (int k=control; k<=100000000; k++){
            double x = tanh(k);
            double val = abs(1-x);
            if (val < pow(10, -n)) {
                cout << k <<", ";
                control=k;
                break;
            }
        }
    }
}

Crossrefs

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

Sequence in context: A365819 A359908 A230872 * A247832 A047368 A020657

Adjacent sequences: A346148 A346149 A346150 * A346152 A346153 A346154

Keyword

nonn

Author

Treanungkur Mal, Jul 07 2021

Status

approved