A004112 -id:A004112 - cp's OEIS frontend

A002485 Numerators of convergents to Pi.

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

Offset: 0

N. J. A. Sloane

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

			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

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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 274.

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

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

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

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

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

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

e=9e9;for(n=1,1e9,abs(tan(n)) 0 monotonically. - M. F. Hasler, Apr 01 2013

Extended and corrected by David Sloan, Sep 23 2002

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.

Original entry on oeis.org

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

N. J. A. Sloane

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

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

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)

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

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

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

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 *)

/* 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

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

A293698 Values of positive integer i such that floor(tan(i)) = 1.

Original entry on oeis.org

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

Offset: 1

V.J. Pohjola, Oct 15 2017

nonn

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

			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.

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

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

rootsp = Flatten[Position[Table[Floor[Tan[i]], {i, 1, 10^3}], 1]]
(* a(n) = rootsp[[n]] *)
(* Alternatively: *)
rootsp = {}; Do[If[Floor[Tan[n]] == 1, AppendTo[rootsp, n]], {n, 1, 10^3}]
rootsp (* a(n) = rootsp[[n]] *)
Select[ Range@ 622, Floor@ Tan@ # == 1 &] (* Robert G. Wilson v, Nov 06 2017 *)

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

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

A382815 Positive numbers k such that abs((sin k)^k) sets a new record.

Original entry on oeis.org

1, 8, 11, 51464, 51819, 52174, 573204, 37362253, 42781604, 122925461, 534483448, 3083975227

Offset: 1

Jwalin Bhatt, Apr 28 2025

			The first few values of abs((sin k)^n), k >= 1, are 0.8414709848, 0.8268218104, 0.002810384737, 0.3280425818, 0.8108146063, 0.0004758860207, 0.05283182049, 0.9179702884, 0.0003429247681, 0.002270688338, 0.9998922779, 0.0005695672234, ... and the record high points are at k = 1, 8, 11, ... - _N. J. A. Sloane_, Apr 28 2025

David Consiglio, Jr., Python using high precision math

Cf. A004112 (analog for |sin n|), A383540.

Module[{x, y, runningMax = 0, positions = {}},
  x = Range[1, 10^6];y = Abs[Sin[x]^x];
  Do[If[y[[i]] > runningMax,runningMax = y[[i]];AppendTo[positions, i];],{i, Length[y]}];
  positions
]

import numpy as np, pandas as pd
x = np.arange(1, 1+10**8)
y = pd.Series(abs(np.sin(x) ** x))
A382815 = sorted([1+int(np.where(y==m)[0][0]) for m in set(y.cummax())])

a(10)-a(12) from David Consiglio, Jr., Apr 28 2025

Definition clarified by Jakub Buczak, May 07 2025

A046956 Numbers k where tan(k) decreases monotonically to 0 (or cot(k) increases).

Original entry on oeis.org

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

Offset: 0

Olivier Gérard

nonn

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

			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

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

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

e=2;for(n=1,1e9, tan(n)>0 && tan(n)M. F. Hasler, Apr 01 2013

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

A307518 Positive integers m where |m*sin(m)| increases to a new record.

Original entry on oeis.org

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

Offset: 1

Alois P. Heinz, Apr 12 2019

nonn

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

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

First differences give A307558.

Cf. A004112, A265735, A272695.

A131975 Numbers n where |sinc(n)| decreases monotonically to 0 (where sinc(x)=sin(x)/x).

Original entry on oeis.org

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

Offset: 1

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

nonn

Cf. A004112, A046947.

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 *)

A131975(nmax)={ local(n=1,aprev=1) ; print1(0) ; while(nA131975(16000) ; \\ R. J. Mathar, Oct 07 2007

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

cp's OEIS Frontend

A002485 Numerators of convergents to Pi.

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Maple

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PARI

PARI

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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.

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A293698 Values of positive integer i such that floor(tan(i)) = 1.

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Mathematica

PARI

PARI

A382815 Positive numbers k such that abs((sin k)^k) sets a new record.

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Mathematica

Python

Extensions

A046956 Numbers k where tan(k) decreases monotonically to 0 (or cot(k) increases).

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A307518 Positive integers m where |m*sin(m)| increases to a new record.

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Author

Keywords

Examples

Links

Crossrefs

A131975 Numbers n where |sinc(n)| decreases monotonically to 0 (where sinc(x)=sin(x)/x).

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Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Extensions