A332095 -id:A332095 - 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

A092328 Solutions of x^2 = ceiling(xrfloor(x/r)) where r=Pi.

Original entry on oeis.org

0, 22, 44, 355, 710, 1065, 1420, 1775, 2130, 2485, 2840, 3195, 312689, 1146408, 5419351, 10838702

Offset: 1

Benoit Cloitre, Feb 14 2004

Does limit n->infinity log(a(n))/n exist?
Notice that the entries above are either numerators of convergents to Pi (A002485) or multiples thereof. - Robert G. Wilson v, Feb 26 2004
a(23) <= 430010946591069243. - Robert G. Wilson v, Jul 19 2019
From M. F. Hasler, Sep 10 2020: (Start)
Appears to be the same as: n >= 0 such that n*tan(n) < 1, cf. A332095. Is there a counterexample?
Most terms are multiples of a smaller term: 44 = 22*2 and a(4..12) = {355, 710, 1065, 1420, 1775, 2130, 2485, 2840, 3195} = 355*{1, 2, 3, ..., 9}. See A332095 for more. (End)

Cf. A249836, A088306, A332095.

Do[ If[ n^2 == Ceiling[n*3.1415926535897932346264*Floor[n/3.1415926535897932346264]], Print[n]], {n, 0, 10^8}] (* Robert G. Wilson v, Feb 26 2004 *)

for(x=0,2000000,if(x^2==ceil(Pi*x*floor(x/Pi)),print1(x,",")))

More terms from Robert G. Wilson v, Feb 26 2004

A337371 Integers k with abs(sin(k)) < 1/k.

Original entry on oeis.org

1, 3, 22, 44, 355, 710, 1065, 1420, 1775, 2130, 2485, 2840, 3195, 312689, 1146408, 5419351, 10838702, 6167950454, 21053343141, 42106686282, 63160029423, 84213372564, 105266715705, 8958937768937, 17917875537874, 428224593349304, 856449186698608, 6134899525417045

Offset: 1

Anian Brosig, Aug 25 2020

nonn

The values > 1 appear to be a subset of the numerators of continued fractions of Pi (A002485) (and/or Pi/2: A096456) and their multiples. Is it possible to find a term k here but not in |A332095| (k |tan k| < 1)? - M. F. Hasler, Oct 09 2020

Cf. A092328, A332095, A046965, A088306, A337249, A332095.

Select[Range[3200], Abs[Sin[#]] < 1/# &] (* Amiram Eldar, Aug 25 2020 *)

print1(1);apply( n-> forstep(n=n,oo,n,abs(sin(n))<1/n||return; print1(","n)), contfracpnqn(c=contfrac(Pi),#c)[1,]); \\ M. F. Hasler, Oct 09 2020

import numpy as np
for x in range(1, 10**9):
    if np.abs(np.sin(x)) < 1/x:
        print(x, end=", ")

More terms from M. F. Hasler, Oct 09 2020

A342171 Nonnegative integers k such that k < sec(k)*csc(k).

Original entry on oeis.org

1, 22, 44, 355, 710, 1065, 1420, 1775, 2130, 2485, 2840, 3195, 260515, 312689, 1146408, 5419351, 10838702, 37362253, 122925461, 534483448, 3083975227, 902209779836, 74357078147863, 214112296674652, 642336890023956, 18190586279576483, 248319196091979065

Offset: 1

José Mauricio de Oliveira Neto, Mar 04 2021

nonn

Conjecture: 2*k/Pi is either a little more than an even integer or a little less than an odd integer.

The conjecture is true. As k > 0 increases, satisfaction of the inequality k < sec(k)*csc(k) requires that sec(k)*csc(k) be a large positive number. Since sec(k)*csc(k) = 1/(sin(k)*cos(k)) = 2/sin(2*k), this requires that sin(2*k) be a small positive number, which occurs only when 2*k/Pi is a little more than an even integer or a little less than an odd integer. - Jon E. Schoenfield, Mar 06 2021

Jon E. Schoenfield, Magma program

Cf. A249836, A332095, A337371.

Select[Range[10^6], # < Sec[#] Csc[#] &] (* Michael De Vlieger, Mar 14 2021 *)

isok(k) = k < 1/(sin(k)*cos(k)); \\ Michel Marcus, Mar 05 2021

import math
i = 1;
while True:
  if(i < 1/(math.cos(i)*math.sin(i))):
    print(str(i) + ", ")
  i += 1

a(22)-a(27) from Jon E. Schoenfield, Mar 06 2021

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A002485 Numerators of convergents to Pi.

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A092328 Solutions of x^2 = ceiling(x*r*floor(x/r)) where r=Pi.

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Mathematica

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A337371 Integers k with abs(sin(k)) < 1/k.

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Mathematica

PARI

Python

Extensions

A342171 Nonnegative integers k such that k < sec(k)*csc(k).

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A092328 Solutions of x^2 = ceiling(xrfloor(x/r)) where r=Pi.