A053873 -id:A053873 - cp's OEIS frontend

A000319 a(n) = floor(b(n)), where b(n) = tan(b(n-1)), b(0)=1.

1, 1, 74, -1, -2, -3, 0, 1, 30, -2, -2, 29, 1, 4, -6, 0, 1, 2, -1, -1, -1, -1, -2, -9, 0, 0, 1, 2, -2, -35, -1, -1, -1, -1, -1, -1, -1, -2, -3, 0, 0, 1, 5, -2, -2, 3, 1, 1, -4, -1, -1, -1, -1, -1, -1, -1, -1, -2, -3, 1, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -2, -3, 0, 1, 2, -1, -2, -21, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

sign

Using 3000-digit precision, interval arithmetic provides an efficient method of computing over 2000000 terms of this sequence. The iteration is stopped when an interval contains an integer. So far, no term equals 319. - T. D. Noe, Mar 07 2008

The question whether 319 occurs is relevant for sequences A053169 and A053873. - Antti Karttunen and M. F. Hasler, Mar 01 2025

			From _José María Grau Ribas_, Apr 13 2010: (Start)
For n=2, tan(tan(1)) = 74.68... (A085665), so a(2)=74.
For n=3, tan(tan(tan(1))) = -0.8635... (A085666), so a(3)=-1. (End)

T. D. Noe, Table of n, a(n) for n = 0..10000
Tim Peters, CSV table of a(n) for n = 0..7366317 in "value,count" format

See A381230 (resp. A381231) for when n (resp. -n) appears.
Cf. A053169, A053873, A085665, A085666.
Cf. A000329 (with round).

Floor[Table[Nest[Tan, 1, n], {n, 1, 200}]] (* José María Grau Ribas, Apr 13 2010 *)

A053169 A paradoxical sequence: a positive integer n is in this sequence if and only if n is not in sequence A_n in the database.

Original entry on oeis.org

4, 7, 9, 11, 12, 13, 15, 17, 18, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 33, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Miklos Szabo (mike(AT)ludens.elte.hu), N. J. A. Sloane, Feb 29 2000

"Not in sequence A_n" means not in the full list of terms, not simply in the list of terms visible in the entry.

The paradox is of course: is 53169 in this sequence?

			Sequence A000001 contains 1, so 1 is not in the sequence; A000002 contains 2, so 2 is not in the sequence; ...

Smullyan, Raymond M, What is the name of this book? : The riddle of Dracula and other logical puzzles, Englewood Cliffs, NJ : Prentice-Hall, c1978, see puzzle 163.

Index entries for sequences whose definition involves A_n (or An).

Cf. A107357.

Complement of A053873.

Thanks to Alexander Hulpke for the information that 19 is in A000019.

Extended to n=100 by N. J. A. Sloane, May 26 2007

A338545 Numbers that escape to infinity when applying this algorithm: if x_i <= rev(x_i) then x_i+1 = rev(x_i) - x_i else x_i+1 = rev(x_i) + x_i.

Original entry on oeis.org

3763, 3853, 3943, 3973, 4492, 4582, 4672, 4762, 4792, 4852, 4882, 4942, 4972, 5194, 5284, 5374, 5464, 5491, 5554, 5581, 5644, 5671, 5734, 5761, 5791, 5824, 5851, 5881, 5914, 5941, 5971, 5993, 6193, 6283, 6373, 6426, 6463, 6490, 6516, 6553

Offset: 1

Ruediger Jehn, Nov 02 2020

All numbers up to 3762 (not already being a palindrome) are mapped sooner or later onto a palindrome and collapse to 0 in the next step of the algorithm. But unlike in the Collatz conjecture (see A006370) where probably all numbers fall back to 1, here most numbers never fall back to 0. Instead, they fall onto repetitive patterns which increase by one order of magnitude every three steps and hence go to infinity. For instance if the algorithm reaches the number 19799999999999991002000000000000088 at step k it will reach 1979999999999999910020000000000000088 at step k+6. And every six steps one 9 is added to the long line of 9's and one 0 is added right of the 2. Up to 100000 there are 3 repetitive patterns, which contain the numbers listed in the second line of the Python code below (escape = [...]).
From the first 1 million numbers, 489806 fall back to zero whereas the other 510194 fall into one of 6 repetitively increasing patterns.
The terms become more and more frequent: up to 10000 they just make 1.78 %, up to 100000 they are 21.88 % and up to 1000000 they are already 51.02 %.
Many numbers (like the 137 in the example below) hit at some point the numbers 8712 and 9801 that are terms of A031877, which is not a coincidence. They fall back to 0.
338545 is a term of this sequence A338545, hence it is also a term of A053873.

			x_0 = 137, x_1 = 731 - 137 = 594, x_2 = 495 + 594 = 1089, x_3 = 9801 - 1089 = 8712, x_4 = 2178 + 8712 = 10890, x_5 = 9801 + 10890 = 20691, x_6 = 19602 + 20691 = 40293, x_7 = 39204 + 40293 = 79497, x_8 = 79497 - 79497 = 0, hence 137 is not a term.

Cf. A031877.

sequence = []
escape = [1090089, 99100089, 9900109899999990109989]
for k in range(1, 100000):
    x = k
    while not x==0:
       if x <= rev(x):
          x = rev(x) - x
       else:
          x = rev(x) + x
       if x in escape:
          x = 0
          sequence.append(k)

A250221 Least k such that A_n(k) = n, or -1 if no such k exists.

Original entry on oeis.org

1, 2, 11, -1, 16, 12, -1, 9, -1, 11, -1, -1, -1, 11, -1, 8, -1, -1, 126, -1, -1, -1, -1, -1, -1, 26, 27, -1, -1, -1, -1, -1, -1, -1, -1, 29, 31, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 33, -1, -1, -1, -1, -1, -1, 19, -1, -1, 45, -1, -1, -1, -1, -1, -1, 35, -1, -1, 8

Offset: 1

Eric Chen, Dec 24 2014

sign

a(A053169(n)) = -1, but what is a(53169)?
a(319) is the first unknown term. (See A000319)
a(241) should be -1. (See A000241)

Cf. A053873, A053169.

A358291 a(n) = smallest k not already in the sequence such that OEIS entry Ak contains n.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 15, 10, 11, 13, 19, 17, 18, 14, 26, 16, 21, 20, 27, 22, 25, 37, 28, 56, 62, 47, 36, 48, 32, 29, 40, 61, 51, 44, 69, 24, 59, 113, 46, 33, 52, 41, 57, 73, 70, 68, 55, 80, 134, 53, 115, 93, 49, 50, 45, 78, 98, 66, 54, 31, 43, 64, 83, 79, 94, 84

Offset: 0

N. J. A. Sloane, Nov 30 2022

			A000001 contains 0, so a(0) = 1.
A000002 contains 1, so a(1) = 2.
k = 10 is the smallest k not yet in the sequence such that Ak = A000010 contains 8, so a(8) = 10.

Index entries for sequences whose definition involves A_n (or An).

Cf. A051070, A053169, A053873.

More terms from Hugo Pfoertner, Dec 01 2022

cp's OEIS Frontend

A000319 a(n) = floor(b(n)), where b(n) = tan(b(n-1)), b(0)=1.

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A053169 A paradoxical sequence: a positive integer n is in this sequence if and only if n is not in sequence A_n in the database.

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A338545 Numbers that escape to infinity when applying this algorithm: if x_i <= rev(x_i) then x_i+1 = rev(x_i) - x_i else x_i+1 = rev(x_i) + x_i.

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Python

A250221 Least k such that A_n(k) = n, or -1 if no such k exists.

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A358291 a(n) = smallest k not already in the sequence such that OEIS entry Ak contains n.

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