A046947 Numbers k such that |sin(k)| (or |tan(k)| or |sec(k)|) decreases monotonically to 0; also |cos(k)| (or |cosec(k)| or |cot(k)|) increases.

1, 3, 22, 333, 355, 103993, 104348, 208341, 312689, 833719, 1146408, 4272943, 5419351, 80143857, 165707065, 245850922, 411557987, 1068966896, 2549491779, 6167950454, 14885392687, 21053343141, 1783366216531, 3587785776203

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

Also numerators of convergents to Pi (A002486 gives denominators) beginning at 1.

Integer circumferences of circles with a(0)=1 and a(n+1) is the smallest integer circumference with corresponding diameter nearer an integer than is the diameter of the circle with circumference a(n). See PARI program. - Rick L. Shepherd, Oct 06 2007

Examples

			|sin(4272943)| = 0.000000549579497810490800503139..., |tan(4272943)| = 0.000000549579497810573797346111..., |sec(4272943)| = 1.00000000000015101881221...
|cos(4272943)| = 0.999999999999848981187793172965367089856..., |cosec(4272943)| = 1819572.97167010734684889..., |cot(4272943)| = 1819572.97166983255709999...
		

References

• K. H. Rosen et al., eds., Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000; p. 293.
• Suggested by a question from Alan Walker (Alan_Walker(AT)sabre.com)

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1946
• Carlos Hernan Lopez Zapata and Nikos Mantzakouras, Diophantine Flint-Hills series: convergence, polylogarithms and pi's bound, ResearchGate (2025). See pp. 6, 69.
• Eric Weisstein's World of Mathematics, Cosecant
• Eric Weisstein's World of Mathematics, Flint Hills Series

Crossrefs

Cf. A004112, A049946. See also A002485, which is the same sequence but begins at 0.

Programs

Maple

Digits := 50; M := 10000; a := [ 1 ]; R := sin(1.); for n from 2 to M do t1 := evalf(sin(n)); if abs(t1)Zerinvary Lajos, Feb 07 2007

Mathematica

z={}; current=1; Do[ If[ Abs[ Sin[ n]] < current, AppendTo[ z, current=Abs[ Sin[ n]]]], {n, 1, 10^7}]; z (* or *)
Join[{1}, Table[ Numerator[ FromContinuedFraction[ ContinuedFraction[Pi, n]]], {n, 1, 23}]] (* Wouter Meeussen *)
Join[{1},Convergents[Pi,30]//Numerator] (* Harvey P. Dale, May 05 2019 *)

PARI

/* Program calculates a(n) without using sin or continued fraction functions */ {d=1/Pi; print1("1, "); for(circum=2,500000000, dm=circum/Pi; dmin=min(dm-floor(dm),ceil(dm)-dm); if(dminRick L. Shepherd, Oct 06 2007

Extensions

More terms from Wouter Meeussen

Further terms from Michel ten Voorde

Edited and extended by Robert G. Wilson v, Jan 28 2003

Typo in examples fixed by Paolo Bonzini, Mar 21 2012