[seqfan] Will these patterns continue for larger numbers?

[Ali Sada]

Hi everyone,

 

If we add a positiveinteger n to and we apply the following algorithm:

“If n iseven, we divide it by two;

If n is odd,we add it to m; where m is the smallest square > n.”

When Icontinued with this algorithm it reached either 1, or 11 as its lowest point.

For example:

13+16=29

29+36=65 

65+81=146 

146/2=73 

73+81=154 

154/2=77 

77+81=158 

158/2=79 

79+81=160

160/32=5

5+9=14

14/2=7

7+9=16

And 16 willgo to 1

The numberof steps before reaching 1 or 11 is: 

8 ,1 ,6 ,2 ,7,7 ,5 ,3 ,17 ,8 ,17 ,8 ,21 ,6 ,21 ,4 ,15 ,18 ,3 ,9 ,13 ,1 ,11 ,9 ,16 ,22 ,16 ,7,20 ,22 ,20 ,5 ,10 ,16 ,8 ,19 ,16 ,4 ,4 ,10 ,16 ,14 ,14 ,2 ,14 ,12 ,12 ,10 ,31,17 ,38 ,23 ,29 ,17 ,27 ,8 ,34 ,21 ,34 ,23 ,15 ,21 ,15 ,6 ,19 ,11 ,19 ,17 ,9 ,9,7 ,20 ,17 ,17 ,7 ,5 ,15 ,5 ,13 ,11 ,58 ,17 ,35 ,15 ,27 ,15 ,57 ,3 ,44 ,15 ,40,13 ,30 ,13 ,47 ,11 ,45 ,32 ,34 ,18,..

 

I wasn’table to check more than 580 numbers because I am using an obsolete software.I checked some big numbers (in the range of 10^6) manually, and couldn’t find differentresults. 

 

I alsochecked when I changed the algorithm slightly (m is the largest square smallerthan n.) The 500 numbers I was able to check always went back to 1 or 5. Starting from n=2, thenumber of steps is:

 

1 ,3 ,2 ,7,4 ,4 ,3 ,6 ,1 ,3 ,5 ,5 ,5 ,7 ,4 ,13 ,7 ,11 ,2 ,23 ,4 ,25 ,6 ,17 ,6 ,8 ,6 ,10 ,8,8 ,5 ,12 ,14 ,10 ,8 ,22 ,12 ,24 ,3 ,16 ,24 ,18 ,5 ,63 ,26 ,20 ,7 ,22 ,18 ,20 ,7,22 ,9 ,9 ,7 ,24 ,11 ,11 ,9 ,11 ,9 ,9 ,6 ,19 ,13 ,13 ,15 ,15 ,11 ,10 ,9 ,21 ,23,23 ,13 ,15 ,25 ,17 ,4 ,62 ,17 ,19 ,25 ,21 ,19 ,27 ,6 ,23 ,64 ,21 ,27 ,29 ,21 ,8,8 ,25 ,23 ,66 ,....

 

 

I gotsimilar results when m was the largest cube, power of 4, etc., but I am less confidentin those results because the software I am using didn’t work well.

 

I wouldreally appreciate it if you could tell me why this is happening.

 

Best,

 

Ali