m|n => a(m)|a(n)

[Christopher Hanusa]

Hello Seqfans,
     I've just come upon a sequence that satisfies the condition "if m 
divides n then a(m) divides a(n)".  (It's related to the number of domino 
tilings of some sequence of regions.)

Two other sequences that satisfy this property are
0)  a(n)=n (trivially)
1)  Fibonacci indexed where f(1)=f(2)=1

Is there anything known about such sequences in general?  or maybe 
not-so-much-in-general?  I realize that there are infinitely many 
sequences that satisfy this --- values for a(p) for p prime are free, and 
a(p^i q^j)=k*lcm(a(p^i q^{j-1}),a(p^{i-1} q^j)), where k is free, etc. But 
maybe there is some theory around that can help me learn more.  How might 
one go about finding a closed form?  Can you think of any other examples? 
What do you think of when you think of this property?

Thanks for any insights,
--Chris