cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

M. F. Hasler, Sep 10 2020

Keywords

Comments

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

Crossrefs

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

Programs

  • PARI
    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

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

Original entry on oeis.org

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

Views

Author

Joseph C. Y. Wong, Aug 21 2020

Keywords

Comments

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

Examples

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

Crossrefs

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

Programs

  • Mathematica
    Select[Range[10^6], Csc[#] > # &] (* Amiram Eldar, Aug 21 2020 *)
  • PARI
    isok(m) = 1/sin(m) > m; \\ Michel Marcus, Aug 27 2020
  • Python
    import math
i = 1
while True:
  if 1 / math.sin(i) > i:
    print(i)
  i += 1

Extensions

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

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

Views

Author

Anian Brosig, Aug 25 2020

Keywords

Comments

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

Crossrefs

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

Programs

  • Mathematica
    Select[Range[3200], Abs[Sin[#]] < 1/# &] (* Amiram Eldar, Aug 25 2020 *)
  • PARI
    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
  • Python
    import numpy as np
for x in range(1, 10**9):
    if np.abs(np.sin(x)) < 1/x:
        print(x, end=", ")

Extensions

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

Views

Author

Joseph C. Y. Wong, Aug 21 2020

Keywords

Comments

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)

Examples

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

Crossrefs

Subsequence of A337371.
Cf. A337249, A092328, A249836.

Programs

  • Mathematica
    Select[Range[10^6], Sec[#] > # &] (* Amiram Eldar, Aug 21 2020 *)
  • PARI
    isok(m) = 1/cos(m) > m; \\ Michel Marcus, Aug 27 2020
  • Python
    import math
i = 1
while True:
  if 1 / math.cos(i) > i:
    print(i)
  i += 1

Extensions

a(7) from Wolfe Padawer, Jan 05 2023
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