[seqfan] List the dividers, sum the digits

[Eric Angelini]

Hello SeqFans,
1) list all dividers of A (1 and A included) ;
2) sum all digits of the above list ;
3) iterate

Example for A=10:
10--> 1,10,2,5 dig.sum=9
 9--> 1,9,3      dig.sum=13
13--> 1,13       dig.sum=14
14--> 1,14,2,7 dig.sum=15
15--> 1,15,3,5 dig.sum=15 (fixed point)

Questions:
1) what are the integers that loop on
themselves like 1 or 15?
2) is there an integer that doesn't
enter in a loop?
3) let's call "a(1)" the smallest integer
looping on itself in 1 stage; "a(2)" the
smallest integer looping on itself in
2 stages; "a(3)" the smallest integer
looping on itself in 3 stages, etc.
What would the "a(n)" sequence be?

My apologizes if this is old hat.
Best,
É.