5x+1

Introduction

The 5x+1 equation multiplies by 5, adds one, and then divides out all the factors or 2 and 3. As the order of this division is significant the detailed descent is a bit ambiguous. In this diagram I draw out the area near the root of the three assuming that the powers of three are divided out first, then the powers of 2.

When compared to the 3x+1 tree there is clearly more scaffolding (yellow) and fewer significant numbers (green). Because we chose to go 6-2-1 the number 3 appears nowhere in the tree. This is true of all odd multiples of three. We could choose to divide out the multiples of 2 first, but this would exclude a separate set of numbers (most multiples of 2).

Here though I only deal in "significant" numbers so the order of division becomes irrelevant. Here is a python script to generate the next significant number in the 5x+1 sequence. It assumes that 'x' isn't a multiple of 2 or 3.

def next(x0):
   x = 5*x0+1
   while x % 2 == 0:
      x /= 2
   while x % 3 == 0:
      x /= 3
   return x

The Tree

This shows the tree generated by seed values up to 301. The labels indicate the number of powers of 2 and 3 that had to be divided out to get from one number to the next. The colour scheme shows the remainder modulo 6 (pink=1, red=5). Click on the image for a larger version.

Size Records

As we did with 3x+1 we can search for seeds that generate the largest value encountered so far, and the longest descent. The following table shows records for seeds less than 1000. The image on the right shows where these values fit into the tree, click on it for more detail.

Seed	Size Record	Length Record	Sequence	Tree
1	1	1	1
7	7	2	7, 1
11	11	3	11, 7, 1
13	13	4	13, 11, 7, 1
17	43		17, 43, 1
23	73	7	23, 29, 73, 61, 17, 43, 1
41	103		41, 103, 43, 1
47		8	47, 59, 37, 31, 13, 11, 7, 1
53	133		53, 133, 37, 31, 13, 11, 7, 1
71	223		71, 89, 223, 31, 13, 11, 7, 1
77	403		77, 193, 161, 403, 7, 1
113		9	113, 283, 59, 37, 31, 13, 11, 7, 1
119	973	16	119, 149, 373, 311, 389, 973, 811, 169, 47, 59, 37, 31, 13, 11, 7, 1
221	1,153		221, 553, 461, 1153, 961, 89, 223, 31, 13, 11, 7, 1
317	1,723		221, 553, 461, 1153, 961, 89, 223, 31, 13, 11, 7, 1
383	45,973	30	383, 479, 599, 749, 1873, 1561, 1301, 3253, 2711, 3389, 8473, 7061, 17653, 14711, 18389, 45973, 38311, 5321, 13303, 5543, 6929, 17323, 401, 1003, 209, 523, 109, 91, 19, 1
919		31	919, 383, 479, 599, 749, 1873, 1561, 1301, 3253, 2711, 3389, 8473, 7061, 17653, 14711, 18389, 45973, 38311, 5321, 13303, 5543, 6929, 17323, 401, 1003, 209, 523, 109, 91, 19, 1

The lists are getting a little long so let's just continue with the values.

Seed	Size Record	Length Record
2,117		34
3,317	57,223	34
4,469	94,723
4,541	111,403
5,081		35
5,933	139,723	35
6,097	139,723
6,473	646,153
6,503		37
10,463		45
12,053		48
28,927		49
31,421	739,903
31,901	1,252,003
39,053		50
44,183	21,532,499
93,727		51
149,957		52
163,733	23,532,223
172,757		55
381,173	26,295,973
414,617		56
447,773		58
468,569	28,733,473
490,007	28,846,153
490,541	35,250,973
519,113	35,980,153

Seed	Size Record	Length Record
593,303	39,477,403
602,591	46,189,903
732,791	52,659,373
733,973	93,767,653
753,023	172,021,483
909,047		60
1,081,877		67
1,274,519	244,236,403
1,463,897	253,659,373
1,597,973	432,643,123
1,993,973		71
2,190,719	965,374,483
2,501,213		72
2,881,397		75
5,744,669	1,433,400,973
6,443,501	2,523,936,373
6,915,353		76
7,747,319	3,087,000,973
7,991,519		85
10,567,001	3,295,827,223
12,447,533	21,220,407,223
23,065,823		87
31,001,117	62,950,442,623	87
46,389,533	83,179,782,223	88
85,689,581	120,037,190,623
88,378,733		89

Glide Records

Glide Records measure the longest encountered sequence of numbers that are all greater than the starting value. <to do>

Working Back

<to do>