A258024 Natural numbers n such that the iteration of the function floor(tan(k)) applied to n eventually reaches [the fixed point] 1 (or any larger integer if such fixed points exist), where k is interpreted as k radians.

1, 4, 23, 26, 45, 48, 67, 70, 89, 92, 105, 111, 114, 121, 127, 133, 136, 143, 149, 155, 158, 171, 177, 180, 183, 193, 199, 202, 205, 215, 221, 224, 227, 243, 246, 249, 265, 268, 271, 290, 293, 300, 312, 315, 334, 337, 344, 356, 359, 378, 381, 400, 403, 422, 425, 444, 447, 460, 466, 469, 476, 482, 488, 491, 498, 504, 510, 513, 526, 532, 535, 538, 548, 554, 557, 560, 570, 576, 579, 582, 598, 601, 604, 620, 623, 626, 645, 648, 655, 667, 670

Offset: 1

Author

V.J. Pohjola, May 16 2015

Keywords

• nonn

Comments

It is stated in the Comments in A000503 that in Floor(tan(n)) "Every integer appears infinitely often. - Charles R Greathouse IV, Aug 06 2012".

It is conjectured that applying the function floor(tan) k times, with k sufficiently large, on the finite sequence floor(tan(n)), n=0...N, the result is a sequence (cf. A258021) composed only of 0’s and 1’s for all values of N.

The original definition was: "Numbers n with property that floor(tan(n)) reduces to 1 (instead of 0) when the function is applied repeatedly to n with deep enough nesting level." If the conjecture above is true, then the new, in theory more inclusive definition produces exactly the same sequence. It has been checked that for at least up to A249836(13) = 1108341089274117551 there are no other strictly positive fixed points beside 1. - Antti Karttunen, May 26 2015

According to Jan Kristian Haugland (cf. link): It is an open problem whether (tan n) > n for infinitely many n, although it has been proved that |tan n| > n for infinitely many n. - Daniel Forgues, May 27 2015

Examples

			For n=0: 0. (0: 0 iteration)
For n=1: 1. (1: 0 iteration) (in this sequence)
For n=2: 2, -3, 0. (0: 2 iterations)
For n=3: 3, -1, -2, 2, -3, 0. (0: 5 iterations)
For n=4: 4, 1. (1: 1 iteration) (in this sequence)
For n=105: 105, 4, 1. (1: 2 iterations) (in this sequence)
For n=3561: 3561, -212, -18, 1. (1: 3 iterations) (in this sequence)
J. K. Haugland found n=37362253 s.t. tan(n) > n. (Cf. link.)
  For n=37362253: 37362253, 37754853, -1, -2, 2, -3, 0. (0: 6 iterations)
Bob Delaney found n=3083975227 s.t. tan(n) > n. (Cf. Robert Israel link.)
  For n=3083975227: 3083975227, 13356993783, -1, -2, 2, -3, 0.
For n s.t. tan(n) > n, see A249836. - _Daniel Forgues_, May 27 2015
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Jan Kristian Haugland, Re: analysis with tan n > n
• Robert Israel, Re: tan n > n

Crossrefs

Disjoint union of A258202 and A258203.

Cf. A258200 (first differences produce an interesting rhythm).

Cf. A258022 (complement provided that function x -> floor(tan(x)) does not form cycles larger than one).

Cf. A000503, A258020, A258021, A249836.

Programs

Mathematica

x = Table[Floor[Tan[n]], {n, 0, 10^4}];
y = NestWhile[Floor[Tan[#]] &, x, UnsameQ, 2];
Flatten[Position[y, 1]] - 1

Extensions

Based on rewording by Daniel Forgues changed the formal definition to include also any hypothetical fixed points larger than one - Antti Karttunen, May 26 2015