A337371 -id:A337371 - cp's OEIS frontend

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

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

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

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

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

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

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Mathematica

PARI

Python

Extensions