A337249 -id:A337249 - cp's OEIS frontend

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

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

A337248 Numbers k for which sec(k) > k.

Original entry on oeis.org

1, 11, 52174, 260515, 37362253, 42781604, 2685575996367

Offset: 1

Joseph C. Y. Wong, Aug 21 2020

This sequence includes abs(m) for many terms m from A088306, including 1, 11, 52174, 260515, 37362253, 42781604, 2685575996367, 65398140378926, 214112296674652, 12055686754159438, 18190586279576483, 1538352035865186794, 1428599129020608582548671, 103177264599407569664999125, 9322105473781932574489648896, .... - Jon E. Schoenfield, Feb 12 2021
From Wolfe Padawer, Jan 05 2023: (Start)
For any given value in this sequence, it is extremely unlikely that it or its negation is not also in A088306. Take the following facts:
[1] |sec(x)| > |tan(x)| for any finite value of sec(x) and tan(x).
[2] |sec(x)| - |tan(x)| approaches 0, and |sec(x)| and |tan(x)| approach infinity, as x approaches (0.5 + n)*Pi where n is any integer.
[3] Any integer k where |sec(k)| > k or |tan(k)| > k must be close to some value of (0.5 + n)*Pi, increasingly so with larger k.
[4] sec(2685575996367) - |tan(2685575996367)| is approximately 8.437*10^-14.
Therefore, for any integer k > 2685575996367 where sec(k) > k, it must be that sec(k) - |tan(k)| < 8.437*10^-14. In order for sec(k) > k but |tan(k)| < k, it must be that k + 8.437*10^-14 > sec(k) > k, a very small interval that only gets smaller as k increases.
It is thus extremely likely, but not yet explicitly proven, that a(8) = 65398140378926, a(9) = 214112296674652, and a(10) = 12055686754159438. Assuming it exists, the smallest k for which sec(k) > k but |tan(k)| < k is probably very large, and it is unknown whether it is currently computable. (End)

			sec(1) = 1.8508... so 1 is a term.

Subsequence of A337371.

Cf. A337249, A092328, A249836.

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

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

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

a(7) from Wolfe Padawer, Jan 05 2023

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

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