7x±1 problem

The

7 x  ±  1

problem concerns the iterated function

     ${\begin{array}{l}\displaystyle {f(n)={\begin{cases}n/2&{\text{if }}n\equiv ~\;\;0{\pmod {2}},\\7n+1&{\text{if }}n\equiv +1{\pmod {4}},\\7n-1&{\text{if }}n\equiv -1{\pmod {4}}.\end{cases}}}\end{array}}$

A317640 The

7 x  ±  1

function:

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, ...}

The question is then whether the orbit of an arbitrary positive integer

n

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

n

is also known as the total stopping time. Orbits are also known as trajectories. The problem is similar to the 3 x + 1 problem.

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 the

7 x  ±  1

function starting with the first few odd numbers.

7 x  ±  1

trajectories

n	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

n

to reach 1 in

7 x  ±  1

problem, or  − 1 if 1 is never reached. (Also called the total stopping time.)

{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, ...}

A318489 Number of steps to reach a lower number than starting value

n

in

7 x  ±  1

problem, or 0 if never reached. (Also called the dropping time, glide, or stopping time.)

{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

• 3 x + 1 problem

References

• David Barina, “7 x  ±  1: Close Relative of Collatz Problem,” 2018; arXiv:1807.00908.
• Keith Matthews’ pages on the 7 x  ±  1 problem.