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7x±1 problem
From OeisWiki
The
problem concerns the iterated function
function:
always converges to the trivial cycle {1, 8, 4, 2, 1, ...} passing through 1. The [total] number of iterations (or [total] number of steps) to reach for the first time the number 1 from
is also known as the total stopping time. Orbits are also known as trajectories. The problem is similar to the 3 x + 1 problem.
function starting with the first few odd numbers.
Trajectory until 1
Steps to 1
(A317753)
1
(1, ...)
0
3
(3, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
13
5
(5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
10
7
(7, 48, 24, 12, 6, 3, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
18
9
(9, 64, 32, 16, 8, 4, 2, 1, ...)
7
11
(11, 76, 38, 19, 132, 66, 33, 232, 116, 58, 29, 204, 102, 51, 356, 178, 89, 624, 312, 156, 78, 39, 272, 136, 68, 34, 17, 120, 60, 30, 15, 104, 52, 26, 13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
53
13
(13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
19
15
(15, 104, 52, 26, 13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
23
17
(17, 120, 60, 30, 15, 104, 52, 26, 13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
27
19
(19, 132, 66, 33, 232, 116, 58, 29, 204, 102, 51, 356, 178, 89, 624, 312, 156, 78, 39, 272, 136, 68, 34, 17, 120, 60, 30, 15, 104, 52, 26, 13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
50
21
(21, 148, 74, 37, 260, 130, 65, 456, 228, 114, 57, 400, 200, 100, 50, 25, 176, 88, 44, 22, 11, 76, 38, 19, 132, 66, 33, 232, 116, 58, 29, 204, 102, 51, 356, 178, 89, 624, 312, 156, 78, 39, 272, 136, 68, 34, 17, 120, 60, 30, 15, 104, 52, 26, 13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
73
23
(23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
16
25
(25, 176, 88, 44, 22, 11, 76, 38, 19, 132, 66, 33, 232, 116, 58, 29, 204, 102, 51, 356, 178, 89, 624, 312, 156, 78, 39, 272, 136, 68, 34, 17, 120, 60, 30, 15, 104, 52, 26, 13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
58
27
(27, 188, 94, 47, 328, 164, 82, 41, 288, 144, 72, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
20
29
(29, 204, 102, 51, 356, 178, 89, 624, 312, 156, 78, 39, 272, 136, 68, 34, 17, 120, 60, 30, 15, 104, 52, 26, 13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
43
A317753 Number of steps for
to reach 1 in
problem, or − 1 if 1 is never reached. (Also called the total stopping time.)
in
problem, or 0 if never reached. (Also called the dropping time, glide, or stopping time.)
7 x ± 1 |
7 x ± 1 |
a (n) = 7 n + 1 if n ≡ +1 (mod 4), a (n) = 7 n − 1 if n ≡ − 1 (mod 4), otherwise a (n) = n / 2. |
- {0, 8, 1, 20, 2, 36, 3, 48, 4, 64, 5, 76, 6, 92, 7, 104, 8, 120, 9, 132, 10, 148, 11, 160, 12, 176, 13, 188, 14, 204, 15, 216, 16, 232, 17, 244, 18, 260, 19, 272, 20, 288, 21, 300, 22, 316, 23, 328, 24, 344, 25, 356, 26, 372, 27, 384, 28, 400, 29, 412, 30, ...}
n |
n |
Every trajectory on negative numbers corresponds to negated trajectory on positive numbers, and vice versa. Therefore, for negative integers, the conjecture is that the sequence will always converge to the cycle passing through − 1. Obviously 0 is stuck on 0.
The following table gives the sequences resulting from iterating the7 x ± 1 |
7 x ± 1 |
n |
(A317753)
n |
7 x ± 1 |
- {0, 1, 13, 2, 10, 14, 18, 3, 7, 11, 53, 15, 19, 19, 23, 4, 27, 8, 50, 12, 73, 54, 16, 16, 58, 20, 20, 20, 43, 24, 24, 5, 47, 28, 325, 9, 70, 51, 32, 13, 13, 74, 272, 55, 55, 17, 17, 17, 276, 59, 40, 21, 40, 21, 21, 21, 63, 44, 63, ...}
n |
7 x ± 1 |
- {0, 1, 12, 1, 8, 1, 4, 1, 4, 1, 42, 1, 8, 1, 4, 1, 4, 1, 23, 1, 20, 1, 4, 1, 4, 1, 12, 1, 16, 1, 4, 1, 4, 1, 282, 1, 12, 1, 4, 1, 4, 1, 229, 1, 50, 1, 4, 1, 4, 1, 8, 1, 35, 1, 4, 1, 4, 1, 8, 1, 50, 1, 4, 1, 4, 1, 46, 1, 8, 1, 4, 1, 4, 1, 225, 1, 8, 1, 4, 1, 4, 1, 35, 1, 16, 1, 4, 1, 4, 1, 46, 1, 27, 1, 4, 1, 4, 1, 16, ...}
See also
References
- David Barina, “7 x ± 1: Close Relative of Collatz Problem,” 2018; arXiv:1807.00908.
- Keith Matthews’ pages on the 7 x ± 1 problem.