A004112 -id:A004112

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A002485

Numerators of convergents to Pi.
(Formerly M3097 N1255)

+10
45

0, 1, 3, 22, 333, 355, 103993, 104348, 208341, 312689, 833719, 1146408, 4272943, 5419351, 80143857, 165707065, 245850922, 411557987, 1068966896, 2549491779, 6167950454, 14885392687, 21053343141, 1783366216531, 3587785776203, 5371151992734, 8958937768937

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

From Alexander R. Povolotsky, Apr 09 2012: (Start)

K. S. Lucas found, by brute-force search, using Maple programming, several different variants of integral identities which relate each of several first Pi convergents (A002485(n)/A002486(n)) to Pi.

I conjecture the following identity below, which represents a generalization of Stephen Lucas's experimentally obtained identities:

(-1)^n*(Pi-A002485(n)/A002486(n)) = (1/abs(i)*2^j)*Integral_{x=0..1} (x^l*(1-x)^m*(k+(k+i)*x^2)/(1+x^2)) dx where {i, j, k, l, m} are some integers (see the Mathematics Stack Exchange link below).

(End)

From a(1)=1 on also: Numbers for which |tan x| decreases monotonically to zero, in the same spirit as A004112, A046947, ... - M. F. Hasler, Apr 01 2013

See also A332095 for n*|tan n| < 1. - M. F. Hasler, Sep 13 2020

REFERENCES

P. Beckmann, A History of Pi. Golem Press, Boulder, CO, 2nd ed., 1971, p. 171 (but beware errors).

CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.

P. Finsler, Über die Faktorenzerlegung natuerlicher Zahlen, Elemente der Mathematik, 2 (1947), 1-11, see p. 7.

K. H. Rosen et al., eds., Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000; p. 293.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Daniel Mondot, Table of n, a(n) for n = 0..1947 (terms 0..201 from T. D. Noe, terms 202..1000 from G. C. Greubel).

E. B. Burger, Diophantine Olympics and World Champions: Polynomials and Primes Down Under, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.

Marc Daumas, Des implantations differentes ..., see p. 8. [Broken link]

Henryk Fuks, Adam Adamandy Kochanski's approximations of Pi: reconstruction of the algorithm, arXiv preprint arXiv:1111.1739 [math.HO], 2011-2014. Math. Intelligencer, Vol. 34 (No. 4), 2012, pp. 40-45.

S. K. Lucas,Integral approximations to Pi with nonnegative integrands

Mathematics Stack Exchange, Is there an integral that proves pi > 333/106

G. P. Michon, Continued Fractions

Eric Weisstein's World of Mathematics, Pi

Eric Weisstein's World of Mathematics, Pi Continued Fraction

Eric Weisstein's World of Mathematics, Pi Approximations

Index entries for sequences related to the number Pi

EXAMPLE

The convergents are 0, 1, 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, 208341/66317, 312689/99532, 833719/265381, 1146408/364913, 4272943/1360120, 5419351/1725033, 80143857/25510582, 165707065/52746197, 245850922/78256779, 411557987/131002976, 1068966896/340262731, 2549491779/811528438, ... = A002485/A002486

MAPLE

Digits := 60: E := Pi; convert(evalf(E), confrac, 50, 'cvgts'): cvgts;

MATHEMATICA

Join[{0, 1}, Numerator @ Convergents[Pi, 29]] (* Jean-François Alcover, Apr 08 2011 *)

PROG

(PARI) contfracpnqn(cf=contfrac(Pi), #cf)[1, ] \\ M. F. Hasler, Apr 01 2013, simplified Oct 13 2020

(PARI) e=9e9; for(n=1, 1e9, abs(tan(n))<e && !print1(n", ") && e=abs(tan(n))) \\ Illustration of |tan a(n)| -> 0 monotonically. - M. F. Hasler, Apr 01 2013

CROSSREFS

Cf. A002486 (denominators), A046947, A072398/A072399.

Cf. A096456 (numerators of convergents to Pi/2).

KEYWORD

nonn,easy,nice,frac

AUTHOR

N. J. A. Sloane

EXTENSIONS

Extended and corrected by David Sloan, Sep 23 2002

STATUS

approved

A046947

+10
18

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

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

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

Eric Weisstein's World of Mathematics, Cosecant

Eric Weisstein's World of Mathematics, Flint Hills Series

EXAMPLE

|sin(4272943)| = 0.000000549579497810490800503139..., |tan(4272943)| = 0.000000549579497810573797346111..., |sec(4272943)| = 1.00000000000015101881221...

|cos(4272943)| = 0.999999999999848981187793172965367089856..., |cosec(4272943)| = 1819572.97167010734684889..., |cot(4272943)| = 1819572.97166983255709999...

MAPLE

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;

with(numtheory): cf := cfrac (Pi, 100): seq(nthnumer(cf, i), i=-1..22 ); # 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 *)

PROG

(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

CROSSREFS

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

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

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

STATUS

approved

A293698

Values of positive integer i such that floor(tan(i)) = 1.

+10
12

1, 4, 23, 26, 45, 48, 67, 70, 89, 92, 111, 114, 133, 136, 155, 158, 177, 180, 183, 199, 202, 205, 221, 224, 227, 243, 246, 249, 265, 268, 271, 290, 293, 312, 315, 334, 337, 356, 359, 378, 381, 400, 403, 422, 425, 444, 447, 466, 469, 488, 491, 510, 513, 532, 535, 538, 554, 557, 560, 576, 579, 582, 598, 601, 604, 620

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The sequence is the first result in the chain of iteration leading to the ultimate sequence A258024.

Sequence terms are also the roots of A000503(i)=1, starting from i=1.

This is a subsequence of A258024 from which this differs for the first time at n=11, where a(11) = 111, while A258024(11) = 105, the term not included in this sequence. Note that A000503(105) = 4, a term which is included in this sequence. - Antti Karttunen, Oct 30 2017

Numbers k such that Pi/4 <= k - m*Pi < arctan(2) for some m. - Robert Israel, Nov 06 2017

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

EXAMPLE

The values of floor(tan(i)), starting from i=0, are given in A000503. Those i, for which floor(tan(i))=1 is true, are the roots of this equation. Thus the roots are the positions of 1 in A000503(i>0).

For n=1, i=1; a(1)=1.

For n=2, i=4; a(2)=4.

For n=3, i=23; a(3)=23.

MATHEMATICA

rootsp = Flatten[Position[Table[Floor[Tan[i]], {i, 1, 10^6}], 1]

(*a(n) = rootsp[[n]]*)

Alternatively:

rootsp = {}; Do[If[Floor[Tan[n]] == 1, AppendTo[rootsp, n]], {n, 1, 10^6}]

rootsp (*a(n) = rootsp[[n]]*)

Select[ Range@ 622, Floor@ Tan@ # == 1 &] (* Robert G. Wilson v, Nov 06 2017 *)

PROG

(PARI) isok(n) = floor(tan(n)) == 1; \\ Michel Marcus, Oct 24 2017

(PARI) first(n) = {my(res = vector(n), i = 0, pi = [Pi, Pi], sols = [atan(1), atan(2)]); while(1, for(j = ceil(sols[1]), floor(sols[2]), i++; if(i>n, return(res)); res[i] = j); sols+=[Pi(), Pi()])} \\ David A. Corneth, Oct 24 2017

CROSSREFS

Cf. A000503, A258024, A293751, A293700, A293701, A293704, A293699, A293702, A293705, A004112, A024814.

KEYWORD

nonn

AUTHOR

V.J. Pohjola, Oct 15 2017

STATUS

approved

A046956

Numbers k where tan(k) decreases monotonically to 0 (or cot(k) increases).

+10
2

1, 4, 7, 10, 13, 16, 19, 22, 355, 104348, 312689, 1146408, 5419351, 85563208, 165707065, 411557987, 1480524883, 2549491779, 8717442233, 14885392687, 35938735828, 56992078969, 78045422110, 99098765251, 120152108392

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

From Jon E. Schoenfield, Aug 10 2006: (Start)

The approach described uses continued fractions containing an even number of terms of which all but the last term are fixed at the values those terms take in the continued fraction for Pi; the final term is initialized at 1 and incremented by 1 each time until it reaches the value taken by that term in the continued fraction for Pi. The semiconvergents and convergents thus obtained are increasingly accurate approximations for Pi, all of which approach Pi from values larger than Pi. Thus the angles whose sizes (in radians) are the numerators of those semiconvergents and convergents approach (from the positive side) integer multiples of Pi, so the tangents of those angles approach zero from positive values.

If we were to use the same approach but with continued fractions having an odd number of terms, i.e., [3] = 3/1; [3;7,i], i=1..15; [3;7,15,1,i], i=1..292; etc., then the semiconvergents and convergents obtained would likewise be increasingly accurate approximations for Pi, but they would approach Pi from values smaller than Pi, so the angles whose sizes (in radians) are the numerators of those semiconvergents and convergents would approach (from the negative side) integer multiples of Pi and thus the tangents of those angles would approach zero from negative values.

Terms after a(0) = 1 are the numerators of the fractions obtained by evaluating all those convergents and semiconvergents of the continued fraction for Pi (A001203) that, as written below, have an even number of partial quotients:

[3;i], i=1..7 (6 semiconvergents and 1 convergent)

[3;7,15,1]

[3;7,15,1,292,1]

[3;7,15,1,292,1,1,1]

[3;7,15,1,292,1,1,1,2,1]

[3;7,15,1,292,1,1,1,2,1,3,1]

[3;7,15,1,292,1,1,1,2,1,3,1,14,i], i=1..2

[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1]

[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,i], i=1..2

[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,i], i=1..2

[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,1,i], i=1..84, etc. (End)

See also A002485 which has a similar property (numerators of convergents to pi = numbers for which |tan a(n)| decreases to zero). - M. F. Hasler, Apr 01 2013

LINKS

Table of n, a(n) for n=0..24.

EXAMPLE

a(1) is the numerator of [3;1] = 3 + 1/1 = 4/1

a(2) is the numerator of [3;2] = 3 + 1/2 = 7/2

...

a(7) is the numerator of [3;7] = 3 + 1/7 = 22/7

a(8) is the numerator of [3;7,15,1] = 3 + 1/(7 + 1/(15 + 1/1)) = 355/113

a(9) is the numerator of [3;7,15,1,292,1] = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + 1/1)))) = 104348/33215

MATHEMATICA

s = Tan[1]; Do[t = Tan[n]; If[t > 0 && t <= s, Print[n]; s = t], {n, 10^9}] (* Ryan Propper, Jul 27 2006 *)

PROG

(PARI) e=2; for(n=1, 1e9, tan(n)>0 && tan(n)<e && !print1(n", ") && e=tan(n)) \\ - M. F. Hasler, Apr 01 2013

CROSSREFS

Cf. A001203, A004112, A002485 (|tan a(n)|->0).

KEYWORD

nonn

AUTHOR

Olivier Gérard

EXTENSIONS

More terms from Michel ten Voorde

2 more terms from Ryan Propper, Jul 27 2006

More terms from Jon E. Schoenfield, Aug 10 2006

Corrected by Don Reble, Nov 20 2006

STATUS

approved

A307518

Positive integers m where |m*sin(m)| increases to a new record.

+10
2

1, 2, 4, 5, 8, 11, 14, 17, 20, 23, 24, 27, 30, 33, 36, 39, 42, 46, 49, 52, 55, 58, 61, 68, 71, 74, 77, 80, 83, 90, 93, 96, 99, 102, 105, 115, 118, 121, 124, 127, 137, 140, 143, 146, 159, 162, 165, 168, 181, 184, 187, 190, 206, 209, 212, 228, 231, 234, 250, 253

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000

EXAMPLE

|a(n)*sin(a(n))|_{n=1..5} = 0.8415..., 1.819..., 3.027..., 4.794..., 7.915... .

CROSSREFS

First differences give A307558.

Cf. A004112, A265735, A272695.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Apr 12 2019

STATUS

approved

A131975

Numbers n where |sinc(n)| decreases monotonically to 0 (where sinc(x)=sin(x)/x).

+10
1

0, 1, 2, 3, 6, 9, 12, 13, 16, 19, 22, 44, 66, 88, 110, 132, 154, 176, 179, 201, 223, 245, 267, 289, 311, 333, 355, 710, 1065, 1420, 1775, 2130, 2485, 2840, 3195, 3550, 3905, 4260, 4615, 4970, 5325, 5680, 6035, 6390, 6745, 7100, 7455, 7810, 8165, 8520, 8875

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..51.

MATHEMATICA

a = {0, 1}; For[n = 2, n < 10000, n++, If[Abs[Sin[n]/n] < Abs[Sin[a[[ -1]]]/a[[ -1]]], AppendTo[a, n]]]; a (* Stefan Steinerberger, Oct 08 2007 *)

PROG

(PARI)

A131975(nmax)={ local(n=1, aprev=1) ; print1(0) ; while(n<nmax, if( abs(sinc(n)) < aprev, print1(", ", n) ; aprev=abs(sinc(n)) ; ) ; n++ ; ) ; }

A131975(16000) ; \\ R. J. Mathar, Oct 07 2007

CROSSREFS

Cf. A004112, A046947.

KEYWORD

nonn,changed

AUTHOR

Laurent A. Guerin (laurent.a.guerin(AT)orange.fr), Oct 06 2007

EXTENSIONS

More terms from R. J. Mathar and Stefan Steinerberger, Oct 07 2007

STATUS

approved

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