A258200 -id:A258200 - cp's OEIS frontend

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

V.J. Pohjola, May 16 2015

nonn

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

			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

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

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.

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

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

A293700 First differences of A293698.

Original entry on oeis.org

3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 3, 16, 3, 3, 16, 3, 3, 16, 3, 3, 16, 3, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 3, 16, 3, 3, 16, 3, 3, 16, 3, 3, 16, 3, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3

Offset: 1

V.J. Pohjola, Oct 16 2017

nonn

Sequence seems to be composed of only three different integers: 3, 16 and 19.
Despite its apparent simplicity, it has interesting palindromic and periodic features and may be conjectured not to be represented in a closed form.
It has a resemblance to the sequences in DNA being composed of four nucleotide bases in varying orders. These sequences, too, contain palindromic substructures having an important role for the genome.
From Robert Israel, Nov 06 2017: (Start)
The only possible values are 3, 16 and 19.
k is in A293698 iff Pi/4 <= k - m*Pi < arctan(2) for some m. We may then verify the following:
If Pi/4 <= k - m*Pi < arctan(2) - 16 + 5*Pi, then k+16 is the next term of A293698.
If arctan(2) - 16 + 5*Pi <= k - m*Pi < 5*Pi/4 - 3, then k+19 is the next term of A293698.
If 5*Pi/4 - 3 <=  k - m*Pi < arctan(2), then k+3 is the next term of A293698. (End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A258200, A258007, A293701, A293704, A293698, A293751, A293699, A293702, A293705.

A293698:= select(i -> floor(tan(i))=1, [$1..1000]):
A293698[2..-1]-A293698[1..-2]; # Robert Israel, Nov 06 2017

rootsp = Flatten[Position[Table[Floor[Tan[i]], {i, 1, 10^6}], 1]];
difp = Differences[rootsp]
(*a(n)=difp[[n]]*)
Differences@ Select[ Range@750, Floor@ Tan@# == 1 &] (* Robert G. Wilson v, Nov 06 2017 *)

lista(nn) = {last = 0; for (n=1, nn, if (floor(tan(n)) == 1, if (last, print1(n-last, ", ")); last = n;););} \\ Michel Marcus, Oct 24 2017

A258007 First differences of A258203.

Original entry on oeis.org

3, 19, 3, 19, 3, 19, 3, 19, 3, 13, 6, 3, 13, 6, 3, 13, 6, 3, 13, 6, 3, 3, 10, 6, 3, 3, 10, 6, 3, 3, 16, 3, 3, 16, 3, 3, 19, 3, 7, 12, 3, 19, 3, 7, 12, 3, 19, 3, 19, 3, 19, 3, 19, 3, 13, 6, 3, 13, 6, 3, 13, 6, 3, 13, 6, 3, 3, 10, 6, 3, 3, 10, 6, 3, 3, 16, 3, 3, 16, 3, 3, 19, 3, 7, 12, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 13, 6

Offset: 1

Antti Karttunen after V.J. Pohjola's amazing A258200, Jun 05 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Differs from A258200 for the first time at n=13.

Cf. A258006, A258203.

(define (A258007 n) (- (A258203 (+ 1 n)) (A258203 n)))

a(n) = A258203(n+1) - A258203(n).

A258006 First differences of A258202.

Original entry on oeis.org

22, 333, 22, 333, 22, 223, 110, 22, 333, 22, 223, 110, 22, 333, 22, 333, 22, 333, 22, 333, 22, 223, 110, 22, 333, 22, 333, 22, 333, 22, 333, 22, 223, 110, 22, 333, 22, 333, 22, 333, 22, 333, 22, 333, 22, 333, 22, 223, 110, 22, 333, 22, 333, 22, 333, 22, 333, 22, 333, 22, 333, 22, 333, 22, 333, 22, 333, 22, 333, 22, 333, 22, 223, 110

Offset: 1

Antti Karttunen, Jun 05 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A258202, A258200, A258007.

(define (A258006 n) (- (A258202 (+ 1 n)) (A258202 n)))

a(n) = A258202(n+1) - A258202(n).

cp's OEIS Frontend

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.

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A293700 First differences of A293698.

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A258007 First differences of A258203.

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A258006 First differences of A258202.

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