[seqfan] Adding digits by couples

[Eric Angelini]

Hello Seqfans (I hope this idea is not old hat),

We change integer abcde...mn (with a,b,c,d,e,..m,n being 
digits) into integer a+b|b+c|c+d|d+e|e+...+m|m+n (where 
the symbol < | > means < concatenate with >. Then we iterate.

So 2012 gives 2+0|0+1|1+2 which is 213.
    213 gives 2+1|1+3 which is 34.
 And 34 gives 3+4 = 7 (which we call "end result")

1) What integers are divisible by their end result?

2) 991 loops -- as 992, 993, 994, 995, 996, 997, 998, 999
   (ex. 991-1810-991...); which integers > 999 also enter 
   in a loop?

3) which is the smallest integer growing for ever (if such
   an integer exists)? I don't know the fate of 1891, for
   example: 1891-91710-10881-18169-99715-181686-9971414...

 Besides, is there an end pattern here: 1991
                                      101810
                                       11991
                                     2101810
                                      311991
                                    42101810
                                     6311991
                                   942101810
Best,                                    ...
É.