A096456 -id:A096456 - 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

A332095 Numbers m such that 0 <= m*tan(m) < 1, ordered by |m|.

Original entry on oeis.org

0, -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

M. F. Hasler, Sep 10 2020

sign

Equivalently, 0 together with integers m such that |tan(m)| < 1/m, multiplied by sign(tan(m)).
The term a(2) = 3 is up to 10^7 the only term m for which tan(m) < 0.
A092328 appears to be a subsequence. Does it contain all terms with tan(m) > 0?
Many 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}.
Indeed, if |m*tan(m)| < 1/k^2 for some k = 1, 2, 3..., then (k*m)*tan(k*m) ~ k^2*m*tan(m) < 1. (E.g., for m = 355, m*tan(m) ~ 0.01.)
The "seeds" for which |m*tan(m)| is particularly small are numerators of convergents of continued fractions for Pi (A002485) (and/or Pi/2: A096456), e.g., a(3) = numerator(22/7), a(5) = numerator(355/113), ...
Other terms in the sequence include: -21053343141*{1, 2, 3, 4, 5}, -8958937768937*{1, 2}, -6134899525417045, -66627445592888887, 430010946591069243, -2646693125139304345*{1, 2, 3, 4, 5}, ...
The absolute values of nonzero terms are a subsequence of A337371. - R. J. Mathar, Sep 24 2020
Can someone find a counterexample for which |sin(m)| < 1/m and |m*tan(m)| > 1? - M. F. Hasler, Oct 09 2020

Cf. A092328, A088306, A337371 (similar, with sin, a superset except for the initial term).

is_A332095(n)={tan(n)*n < 1 && n*tan(n) >= 0}
for(n=0,oo, n*abs(tan(n))<1 && print1(sign(tan(n))*n", "))
/* Much faster: apply to numerators of convergents of Pi the function check(n) which prints all nonzero k*n in the sequence and returns the largest such k, largest in magnitude, possibly negative. N.B.: stops when (k+1)n is not in the sequence, so e.g., n = 11 (in convergents of Pi/2) does not give 22 and 44! */
print1(0); apply( {check(n)=for(i=1,oo,abs(i*n*tan(i*n))<1||return(sign(tan(n))*(i-1)); print1(", "sign(tan(n*i))*i*n))}, contfracpnqn(c=contfrac(Pi),#c)[1,]) \\ M. F. Hasler, Oct 09 2020

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

cp's OEIS Frontend

A096464 Let p(k)/q(k) = A096456(k)/A096463(k) be the k-th convergent to Pi/2; sequence gives numbers n such that |tan(p(n))|/p(n) sets a new maximal record.

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A002485 Numerators of convergents to Pi.

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Maple

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PARI

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A332095 Numbers m such that 0 <= m*tan(m) < 1, ordered by |m|.

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PARI

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

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A096463 Denominators of convergents to Pi/2.

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A346033 a(n) is the smallest integer k > 0 such that 10^(-n-1) < |sin(k) - round(sin(k))| < 10^(-n).

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