A265735 -id:A265735 - cp's OEIS frontend

A337249 Numbers k for which csc(k) > k.

1, 3, 44, 710, 1420, 2130, 2840, 312689, 10838702, 6167950454, 21053343141, 63160029423, 105266715705

Offset: 1

Joseph C. Y. Wong, Aug 21 2020

a(14) > 1.129*10^12, if it exists. - Kevin P. Thompson, Nov 07 2021

a(14) exists. The numbers 428224593349304, 6134899525417045, 66627445592888887, 430010946591069243, and 2646693125139304345 all satisfy csc(k) > k and are larger than a(13). It is not yet proven whether these are a(14) - a(18) or if there are any other numbers in the sequence before or between them. - Wolfe Padawer, Apr 11 2023

			csc(1) = 1.1884... so 1 is a term.

Subsequence of A080142, A046955.
Subsequence of A265735 and A325158 if you omit the first term of A337249.
Cf. A092328, A337248.

Select[Range[10^6], Csc[#] > # &] (* Amiram Eldar, Aug 21 2020 *)

isok(m) = 1/sin(m) > m; \\ Michel Marcus, Aug 27 2020

import math
i = 1
while True:
  if 1 / math.sin(i) > i:
    print(i)
  i += 1

a(11)-a(13) from Kevin P. Thompson, Nov 07 2021

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.

A265739 Numbers k such that there exists at least one integer in the interval [Pik - 1/k, Pik + 1/k].

Original entry on oeis.org

1, 2, 6, 7, 14, 21, 28, 106, 113, 226, 339, 452, 565, 678, 791, 904, 1017, 1130, 1243, 1356, 1469, 1582, 1695, 1808, 1921, 33102, 33215, 66317, 99532, 165849, 265381, 364913, 729826, 1360120, 1725033, 3450066, 5175099, 25510582, 27235615

Offset: 1

Michel Lagneau, Dec 15 2015

nonn

Conjecture: the sequence is infinite.
See the reference for a similar problem with Fibonacci numbers.
For k > 1, the interval [Pi*k - 1/k, Pi*k + 1/k] contains exactly one integer.
The corresponding integers in the interval [Pi*k - 1/k, Pi*k + 1/k] are 3, 4, 6, 19, 22, 44, 66, 88, ... (see A265735).
The sequence is infinite by Dirichlet's approximation theorem. In other words, the irrationality measure of Pi is at least 2 so this sequence is infinite. - Charles R Greathouse IV, Nov 07 2022

			For k=1, there exist two integers, 3 and 4, in the interval [1*Pi - 1/1, 1*Pi + 1/1] = [2.14159..., 4.14159...];
for k=2, the number 6 is in the interval [2*Pi - 1/2, 2*Pi + 1/2] = [5.783185..., 6.783185...].
for k=6, the number 19 is in the interval [6*Pi - 1/6, 6*Pi + 1/6] = [18.682889..., 19.016223...].

Takao Komatsu, The interval associated with a Fibonacci number, The Fibonacci Quarterly, Volume 41, Number 1, February 2003.
Eric Weisstein's World of Mathematics, Pi Continued Fraction

Cf. A000796, A265735. Contains A002486 (without the first two terms) as a subsequence.

# program gives the interval [a,b], the first integer in [a,b] and n
nn:=10^9:
for n from 1 to nn do:
x1:=evalhf(Pi*n-1/n):y1:=evalhf(Pi*n+1/n):
x:=floor(x1):y:=floor(y1):
for j from x+1 to y do:
printf("%g %g %d %d\n",x1,y1,j,n):
od:
od:

A362602 Integers in the interval [Pik - 1/k, Pik + 1/k] for some k > 0 that are numerators of convergents to 2*Pi.

Original entry on oeis.org

6, 19, 44, 333, 710, 103993, 312689, 2292816, 4272943, 10838702, 80143857, 411557987, 2549491779, 14885392687, 42106686282, 1783366216531, 8958937768937, 288469374822515, 856449186698608, 5706674932067741, 12269799050834090, 30246273033735921, 133254891185777774

Offset: 1

Alexander R. Povolotsky and Stefano Spezia, Apr 27 2023

nonn

Conjecture: the sequence is infinite.

Intersection of A046955 and A265735.

hmax=30; A046955=Join[{1}, Numerator[Convergents[2Pi, hmax]]]; a={}; For[h=2, h<=hmax, h++, k=Intersection[List[Floor[Last[x/.N[Solve[Pi*x^2 - 1 - Part[A046955,h] x == 0, x], 2*hmax]]]], List[Ceiling[Last[x/.N[Solve[Pi*x^2 + 1 - Part[A046955,h] x == 0, x], 2*hmax]]]]]; If[k!={} && Ceiling[k*Pi - 1/k] == Floor[k*Pi + 1/k], AppendTo[a,Part[A046955,h]]]]; a

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A337249 Numbers k for which csc(k) > k.

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

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A265739 Numbers k such that there exists at least one integer in the interval [Pik - 1/k, Pik + 1/k].

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Maple

A362602 Integers in the interval [Pik - 1/k, Pik + 1/k] for some k > 0 that are numerators of convergents to 2*Pi.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

cp's OEIS Frontend

A337249 Numbers k for which csc(k) > k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

A265739 Numbers k such that there exists at least one integer in the interval [Pi*k - 1/k, Pi*k + 1/k].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A362602 Integers in the interval [Pi*k - 1/k, Pi*k + 1/k] for some k > 0 that are numerators of convergents to 2*Pi.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A265739 Numbers k such that there exists at least one integer in the interval [Pik - 1/k, Pik + 1/k].

A362602 Integers in the interval [Pik - 1/k, Pik + 1/k] for some k > 0 that are numerators of convergents to 2*Pi.