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

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

Offset: 1

Author

Joseph C. Y. Wong, Aug 21 2020

Keywords

• nonn
• more

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