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

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

Author

José Mauricio de Oliveira Neto, Mar 04 2021

Keywords

• nonn

Comments

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

Links

• Jon E. Schoenfield, Magma program

Crossrefs

Cf. A249836, A332095, A337371.

Programs

Mathematica

Select[Range[10^6], # < Sec[#] Csc[#] &] (* Michael De Vlieger, Mar 14 2021 *)

PARI

isok(k) = k < 1/(sin(k)*cos(k)); \\ Michel Marcus, Mar 05 2021

Python

import math
i = 1;
while True:
  if(i < 1/(math.cos(i)*math.sin(i))):
    print(str(i) + ", ")
  i += 1

Extensions

a(22)-a(27) from Jon E. Schoenfield, Mar 06 2021