[seqfan] Sequence related to Collatz 3n+1

[DAN_CYN_J]

Hi all seq. fans. 
  
In Collatz 3n+1 conjecture the first and following 
seeds that have an integer divisor of its total 
sequence sum. This sum and level (sequence count) 
includes the seed. 

This unique seed sequence with this property = 
1,57,847,1694,3039,3388,3479,6078,6776,6958,13916,27832,55664,111328,236107,246721  ... 

  

This sequence is not in OEIS. 

  

Explaining it further below. 
 1,4,2,1 = level 4,seed 1 and sum 8 of total sequence and divisor 8.  (8*1)=sum 8 
 57..4,2,1 =level 33,seed 57 and sum 1653 of total sequence and divisor 29. (29*57)= sum 1653 
 847..4,2,1 =level 34,seed 847 and sum 49126 and divisor 58. (58*847)= sum 49126 
 etc. 

The associated table for each seed. 

Level |seed     | sum        | divisor 
  4      | 1           |  8            | 8 
----------------------------- 
 33    | 57        |  1653       | 29 
----------------------------- 
 34   | 847      | 49126      | 58 
----------------------------- 
 35   | 1694   | 50820      | 30 
----------------------------- 
 155  | 3039   | 449772   | 148 
----------------------------- 
 36   | 3388   | 54208     | 16 
----------------------------- 
 57   | 3479   | 118286   |  34 
----------------------------- 
 156  | 6078   | 455850   |  75 
----------------------------- 
 37   | 6776   | 60984     |  9 
----------------------------- 
 58   | 6958   | 125244    |  18 
----------------------------- 
 59   | 13916  | 139160    |  10 
----------------------------- 
 60   | 27832  | 166992    |  6 
----------------------------- 
 61   | 55664  | 222656    |  4 
----------------------------- 
 62   | 111328 | 333984   |  3 
----------------------------- 
 50   | 236107 | 4958247  |  21 
----------------------------- 
 151  | 246721 | 4440978  |  18 
----------------------------- 
 172  | 311257 | 5602626  |  18 
----------------------------- 
 152  | 493442 | 4934420  |  10 
----------------------------- 
 173  | 622514 | 6225140  |  10 
----------------------------- 
 153  | 986884 | 5921304  |  6 
----------------------------- 
 174  |1245028 | 7470168  |  6 
----------------------------- 
 143  |1328233 | 63755184 | 48 
----------------------------- 
 154  |1973768 | 7895072  |  4 
----------------------------- 
 190  |2052521 | 340718486|  166 
------------------------------ 
 175  |2490056 | 9960224   |  4 
------------------------------ 
 144  |2656466 | 66411650  | 25 
------------------------------ 
 155  |3947536 | 11842608  | 3 
------------------------------ 
 191  |4105042 | 344823528 | 84 
------------------------------ 
 176  |4980112 | 14940336  | 3 
------------------------------ 
 192  |8210084 | 353033612 | 43 
... 
The sums and levels and divisors are all over the place 
within this table but the associated seeds are in sequence. 
So far there are only two discrete prime divisors of the sums. (3,29). 
There is only one duplicate of a level so far =(155) 

Some of the levels that are in sequence are also 
interesting such as  level 57,58,59,60,61,62. 
The associated divisors are 34,18,10,6,4,3 
The gap in these divisors are 16,8,4,2,1 giving the 
Collatz terminating sequence. 
Another level sequence that is also interesting 
(151,152,153,154,155) 
with gaps  = (8,4,2,1) of their associated divisors =(18,10,6,4,3) 
Another is -- 
(172,173,174,175,176) 

  

It appears there will be many of these sequential level sets 
but I am not sure if the limit will only be (6) as with levels (57->62). 
If any divisor is odd that ends that level sequence or there 
is only one level with that one odd divisor. 

  

The gaps between seeds get larger and larger in this discrete sequence. 

  

Probably not interesting enough for OEIS? 



Dan