[seqfan] Chasing base-10 Harshads

[Eric Angelini]

« A Harshad number, or Niven number (in a given number base), 
  is an integer that is divisible by the sum of its digits
  (when written in that base). » [Wikipedia]

Hello Seqfans,
Let us start with 11; is 11 divisible by (1+1)=2?
No. We then add 2 to 11 => 13

Is 13 divisible by (1+3)=4?
No. We then add 4 to 13 => 17

Is 17 divisible by (1+7)=8?
No. We then add 8 to 17 => 25
...

Non-Harshad 11 needs 25 steps to hit 247 -- which is Harshad:

11-13-17-25-32-37-47-58-71-79-95-109-119-130-134-142-149-163-
173-184-197-214-221-226-236-247 (247/13=19)

n   steps to reach a Harshad:
1     0
2     0
3     0
4     0
5     0
6     0
7     0
8     0
9     0
10    0
11    25
12    0
13    24
14    4
15    1
16    6
17    23
18    0
19    2
20    0
21    0
...
0-step are the Harshad numbers, of course:
http://www.research.att.com/~njas/sequences/A005349

We could build a seq where n is the required number of steps
for the smallest a(n) to hit a Harshad; this seq would start
like this (I think):

S = 15,19,a,14,b,16,...

15 is the smallest integer needing 1 step  to hit a Harshad
19 is the smallest integer needing 2 steps to hit a Harshad
a  is the smallest integer needing 3 steps to hit a Harshad
14 is the smallest integer needing 4 steps to hit a Harshad
b  is the smallest integer needing 5 steps to hit a Harshad
16 is the smallest integer needing 6 steps to hit a Harshad
...

Could someone compute a hundred or so terms of S (if of interest)?
Is it possible for an integer not to hit an Harshad at some point?

Best,
É.