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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.

%I A046947 #50 Feb 16 2025 08:32:39
%S A046947 1,3,22,333,355,103993,104348,208341,312689,833719,1146408,4272943,
%T A046947 5419351,80143857,165707065,245850922,411557987,1068966896,2549491779,
%U A046947 6167950454,14885392687,21053343141,1783366216531,3587785776203
%N 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.
%C A046947 Also numerators of convergents to Pi (A002486 gives denominators) beginning at 1.
%C A046947 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
%D A046947 K. H. Rosen et al., eds., Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000; p. 293.
%D A046947 Suggested by a question from Alan Walker (Alan_Walker(AT)sabre.com)
%H A046947 Michael De Vlieger, <a href="/A046947/b046947.txt">Table of n, a(n) for n = 0..1946</a>
%H A046947 Carlos Hernan Lopez Zapata and Nikos Mantzakouras, <a href="https://doi.org/10.13140/RG.2.2.13412.90240">Diophantine Flint-Hills series: convergence, polylogarithms and pi's bound</a>, ResearchGate (2025). See pp. 6, 69.
%H A046947 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Cosecant.html">Cosecant</a>
%H A046947 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FlintHillsSeries.html">Flint Hills Series</a>
%e A046947 |sin(4272943)| = 0.000000549579497810490800503139..., |tan(4272943)| = 0.000000549579497810573797346111..., |sec(4272943)| = 1.00000000000015101881221...
%e A046947 |cos(4272943)| = 0.999999999999848981187793172965367089856..., |cosec(4272943)| = 1819572.97167010734684889..., |cot(4272943)| = 1819572.97166983255709999...
%p A046947 Digits := 50; M := 10000; a := [ 1 ]; R := sin(1.); for n from 2 to M do t1 := evalf(sin(n)); if abs(t1)<R then R := abs(t1); a := [ op(a), n ]; fi; od: a;
%p A046947 with(numtheory): cf := cfrac (Pi,100): seq(nthnumer(cf,i), i=-1..22 ); # _Zerinvary Lajos_, Feb 07 2007
%t A046947 z={}; current=1; Do[ If[ Abs[ Sin[ n]] < current, AppendTo[ z, current=Abs[ Sin[ n]]]], {n, 1, 10^7}]; z (* or *)
%t A046947 Join[{1}, Table[ Numerator[ FromContinuedFraction[ ContinuedFraction[Pi, n]]], {n, 1, 23}]] (* _Wouter Meeussen_ *)
%t A046947 Join[{1},Convergents[Pi,30]//Numerator] (* _Harvey P. Dale_, May 05 2019 *)
%o A046947 (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(dmin<d, print1(circum,", "); d=dmin))} /* or could use dmin=min(frac(dm),1-frac(dm)) above */ \\ _Rick L. Shepherd_, Oct 06 2007
%Y A046947 Cf. A004112, A049946. See also A002485, which is the same sequence but begins at 0.
%K A046947 nonn,nice
%O A046947 0,2
%A A046947 _N. J. A. Sloane_
%E A046947 More terms from _Wouter Meeussen_
%E A046947 Further terms from _Michel ten Voorde_
%E A046947 Edited and extended by _Robert G. Wilson v_, Jan 28 2003
%E A046947 Typo in examples fixed by _Paolo Bonzini_, Mar 21 2012