[seqfan] a(n) divides the concatenation

[Eric Angelini]

Hello SeqFans (again, sorry to invade, last post of April First);

We look for the smallest integer a(n) 
no yet present in S such that a(n) divides
the concatenation of a(n-1) and a(n).
S starts with a(1)=1

S is infinite, of course, and seems to
look like this (hope no bugs were introduced 
by hand):

S=1,2,4,5,10,20,8,16,25,50,40,32,64,80,100,125,...

(we see for example here that 2 divides 12, that 4 divs 24, that 5 divs 45, that 10 divs 510, etc.)

Is there a pattern somewhere? Will the integer "3" appear at some point? 
Is S interesting?

Best,
É.