[seqfan] Summer sequence boot camp : adding digits to the right... and more

[Eric Angelini]

Hello SeqFans,
Consider this way of producing integers:
- start with integer A -- ex. 2056
- add the leftmost digit to its neighbor : 2+0=2
- add the next leftmost digit to its neighbor on the 
  right : 0+5=5
- proceed like this until the rightmost digit of A is 
  involved in an addition : 5+6=11
- concatenate the partial sums computed so far to 
  produce integer B : [2][5][11] --> 2511

So 2056 produces 2511.

Let's say that 2056 is the predecessor of 2511.
***Question (1):
   What are the first 100 integers with no predecessors?
(0 starts the sequence and 110 is the next term, I guess)
(note that some integers have more than one predecessor:
 "9" has predecessors 18, 27, 36, 45, 54, 63, 72, 81 and 90)
----------
One might be interested in integers "which loop":
991 --> 1810 --> 991
***Question (2)
   What are the first 100 "looping integers"?
----------
"Fibonacci in digits" by Bodo Zinser echoes this way of
producing integers: 
http://oeis.org/A093086

Any integer having at least two digits can start such a
"Zinser-sequence". For instance:
10 --> 1, 0, 1, 1, 2, 3, 5, 8, 1, 3, 9, 4, 1, 2, 1, 3, 5, 3...
11 --> 1, 1, 2, 3, 5, 8, 1, 3, 9, 4, 1, 2, 1, 3, 5, 3, 3, 4...
12 --> 1, 2, 3, 5, 8, 1, 3, 9, 4, 1, 2, 1, 3, 5, 3, 3, 4, 8...
(these first three sequences quickly merge with A093086 -- as
will do n=23, n=35 and n=58, of course)
13 --> 1, 3, 4, 7, 1, 1, 8, 2, 9, 1, 0, 1, 1, 1, 0, 1, 1, 2...
14 --> 1, 4, 5, 9, 1, 4, 1, 0, 5, 5, 1, 5, 1, 0, 6, 6, 6, 1...
...
2011 --> 2,0,1,1,2,1,2,3,3,3,5,6,6,8,1,1,1,2,1,4,9,2,2,3,...
Bodo Zinser suggests that the "Zinser-10" digits are not 
periodic.
***Question (3)
   Is there a "Zinser-n" sequence with digit periodicity?
----------
Instead of adding, we could multiply digits in the same way.
10 --> 1, 0, 0, 0, 0, 0,...
11 --> 1, 1, 1, 1, 1, 1,...
12 --> 1, 2, 2, 4, 8, 3, 2, 2, 4, 6, 4, 8, 2, 4, 2, 4, 3, 2...
(this is http://oeis.org/A045777 by Erich Friedman)
13 --> 1, 3, 3, 9, 2, 7, 1, 8, 1, 4, 7, 8, 8, 4, 2, 8, 5, 6...
(this is http://oeis.org/A060310 by Jason Earls)
14 --> 1, 4, 4, 1, 6, 4, 6, 2, 4, 2, 4, 1, 2, 8, 8, 8, 4, 2...
...
2011 --> 2,0,1,1,0,0,1,0,0,0,0,0,0,0,...
***Question (4):
   What are the integers _not_ producing an infinite trail of 
   zeroes ("2011" produces such a trail -- as "13" will probably
   do, because of the presence of a 5 in position 17 of Erich's
   A045777)

If we play with integers producing other integers with the 
"multiply" law instead of the "add" law, we could ask ourselves
the same questions as above about predecessors and loops.
Here is an example of an integer-chain, starting with 193, and
obeying the "multiply" law:

193-927-1814-884-6432-24126-84212-32822-...

***Question (5):
   Are there "looping integers"?

***Question (6):
   What are the first 100 integers with no predecessors?
(0 starts the sequence and 101 is the next term, I guess)
Best,
É.