%I #74 Feb 04 2019 14:23:58

%S 1,4,23,26,45,48,67,70,89,92,105,111,114,121,127,133,136,143,149,155,

%T 158,171,177,180,183,193,199,202,205,215,221,224,227,243,246,249,265,

%U 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

%N 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.

%C 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".

%C 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.

%C 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

%C 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

%H Antti Karttunen, <a href="/A258024/b258024.txt">Table of n, a(n) for n = 1..10000</a>

%H Jan Kristian Haugland, <a href="http://sci.tech-archive.net/Archive/sci.math/2007-03/msg00666.html">Re: analysis with tan n > n</a>

%H Robert Israel, <a href="http://sci.tech-archive.net/Archive/sci.math/2009-03/msg01170.html">Re: tan n > n</a>

%e For n=0: 0. (0: 0 iteration)

%e For n=1: 1. (1: 0 iteration) (in this sequence)

%e For n=2: 2, -3, 0. (0: 2 iterations)

%e For n=3: 3, -1, -2, 2, -3, 0. (0: 5 iterations)

%e For n=4: 4, 1. (1: 1 iteration) (in this sequence)

%e For n=105: 105, 4, 1. (1: 2 iterations) (in this sequence)

%e For n=3561: 3561, -212, -18, 1. (1: 3 iterations) (in this sequence)

%e J. K. Haugland found n=37362253 s.t. tan(n) > n. (Cf. link.)

%e For n=37362253: 37362253, 37754853, -1, -2, 2, -3, 0. (0: 6 iterations)

%e Bob Delaney found n=3083975227 s.t. tan(n) > n. (Cf. Robert Israel link.)

%e For n=3083975227: 3083975227, 13356993783, -1, -2, 2, -3, 0.

%e For n s.t. tan(n) > n, see A249836. - _Daniel Forgues_, May 27 2015

%t x = Table[Floor[Tan[n]], {n, 0, 10^4}];

%t y = NestWhile[Floor[Tan[#]] &, x, UnsameQ, 2];

%t Flatten[Position[y, 1]] - 1

%o (Scheme, with Antti Karttunen's IntSeq-library)

%o (define A258024 (MATCHING-POS 1 0 (lambda (n) (> (A258021 n) 0))))

%o ;; _Antti Karttunen_, May 24 2015

%Y Disjoint union of A258202 and A258203.

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

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

%Y Cf. A000503, A258020, A258021, A249836.

%K nonn

%O 1,2

%A _V.J. Pohjola_, May 16 2015

%E 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