The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A343515 a(n) is the number of real solutions to the equation sin(x) = x/n. 0
 1, 3, 3, 3, 3, 3, 3, 7, 7, 7, 7, 7, 7, 7, 11, 11, 11, 11, 11, 11, 15, 15, 15, 15, 15, 15, 19, 19, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 27, 27, 27, 27, 27, 27, 31, 31, 31, 31, 31, 31, 35, 35, 35, 35, 35, 35, 35, 39, 39, 39, 39, 39, 39, 43, 43, 43, 43, 43 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS All terms are odd. All terms are congruent to 3 modulo 4 after the first term. Proof: define sin(x)/x to be 1 at x = 0. If a(n) == 1 (mod 4), then the horizontal line y = 1/n is tangent to the curve y = sin(x)/x at (x_n, sin(x_n)/x_n) for some x_n >= 0. We have tan(x_n) = x_n and sin(x_n)/x_n = 1/n, so cos(x_n) = 1/n. By the Lindemann-Weierstrass theorem, we have either x_n = 0 or x_n must be transcendental (if x is a nonzero algebraic number, then exp(x) is transcendental). On the other hand, x_n = tan(x_n) = sqrt(n^2-1) is algebraic, so the only possibility is n = 1. - Jianing Song, Jul 13 2021 LINKS FORMULA a(n) ~ 4*(floor((n-Pi/2)/(2*Pi))+1)-1. For n > 1, a(n) = 4*(floor((n-Pi/2)/(2*Pi))+1)-1 + r(n), where r(n) is an error term defined as follows: let E be the system of equations given by cos(sqrt(n^2-1)) = 1/n and sin(sqrt(n^2-1)) = sqrt(n^2-1)/n; r(n) = 4 if the closest solution of E from the left to Pi/2 + 2*Pi*(floor((n-Pi/2)/2*Pi)+1) is smaller than n; r(n) = 0 otherwise. From Jianing Song, Jul 13 2021: (Start) Define x_k to be root of tan(x) = x in [k*Pi, (k+1)*Pi), k >= 0. For n > 1, if sec(x_(2*k)) < n < sec(x_(2*k+2)) (or equivalently, x_(2*k) < sqrt(n^2-1) < x_(2*k+2)), then a(n) = 4*k + 3. For n >= 2, a(n+1) - a(n) is either 0 or 4. a(n+1) - a(n) = 4 if n is of the form floor(sec(x_(2*k))) = floor(sqrt((x_(2*k))^2+1)) for some k > 0. (End) EXAMPLE a(3) = 3 because the equation sin(x) = x/3 has 3 real solutions: {-2.27886..., 0, 2.27886...}. MATHEMATICA Join[{1}, Table[CountRoots[n*Sin[x] - x, {x, -n, n}], {n, 2, 100}]] (* Vaclav Kotesovec, Jun 25 2021 *) CROSSREFS Sequence in context: A111233 A210746 A283986 * A351836 A105159 A334625 Adjacent sequences: A343512 A343513 A343514 * A343516 A343517 A343518 KEYWORD nonn AUTHOR Pablo Hueso Merino, Apr 17 2021 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 1 21:09 EST 2022. Contains 358484 sequences. (Running on oeis4.)