A213082 Values of n for which the number of roots of the function sin(x)/x - 1/n increases.

1, 2, 8, 15, 21, 27, 33, 40, 46, 52, 59, 65, 71, 77, 84, 90, 96, 103, 109, 115, 121, 128, 134, 140, 147, 153, 159, 165, 172, 178, 184, 191, 197, 203, 209, 216, 222, 228, 235, 241, 247, 253, 260, 266, 272, 279, 285, 291, 297, 304, 310, 316, 323, 329, 335

Offset: 1

Author

Gordon Roesler, Jul 22 2012

Keywords

• nonn

Comments

Maxima of sin(x)/x correspond to odd solutions of x(m)=tan(x(m)). At maxima, sin(x(m))/x(m)=sin(tan(x(m)))/tan(x(m)). Number of roots of f(x)=sin(x)/x - 1/n increases when n = int(x(m)/sin(x(m))+1.

Examples

			For n=1 there is 1 root, for n=2,...7 there are 2 roots, for n=8,...14 there are 6 roots, etc.
		

Programs

Mathematica

t = Table[x*Cos[x] - Sin[x], {x, 400}]; t2 = {1, 2}; Do[If[t[[n]] > 0 && t[[n + 1]] < 0, AppendTo[t2, n + 1]], {n, Length[t] - 1}]; t2 (* T. D. Noe, Jul 30 2012 *)

Extensions

Terms after a(8) from T. D. Noe, Jul 30 2012