cp's OEIS Frontend

A343515 a(n) is the number of real solutions to the equation sin(x) = x/n.

%I A343515 #32 Jul 15 2021 15:08:53
%S A343515 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,
%T A343515 19,19,19,19,19,23,23,23,23,23,23,23,27,27,27,27,27,27,31,31,31,31,31,
%U A343515 31,35,35,35,35,35,35,35,39,39,39,39,39,39,43,43,43,43,43
%N A343515 a(n) is the number of real solutions to the equation sin(x) = x/n.
%C A343515 All terms are odd.
%C A343515 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
%F A343515 a(n) ~ 4*(floor((n-Pi/2)/(2*Pi))+1)-1.
%F A343515 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.
%F A343515 From _Jianing Song_, Jul 13 2021: (Start)
%F A343515 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.
%F A343515 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)
%e A343515 a(3) = 3 because the equation sin(x) = x/3 has 3 real solutions: {-2.27886..., 0, 2.27886...}.
%t A343515 Join[{1}, Table[CountRoots[n*Sin[x] - x, {x, -n, n}], {n, 2, 100}]] (* _Vaclav Kotesovec_, Jun 25 2021 *)
%K A343515 nonn
%O A343515 1,2
%A A343515 _Pablo Hueso Merino_, Apr 17 2021