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

[David Harden]

>>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?

One thing I think of is that the sequence defined by a_0=0, a_(n+1)=p(a_n), where p(x) is in Z[x], has this property.
Another thing I think of (which I haven't explored much) is the related sequence {b_n}, where b_n = product( (a_d)^mu(n/d), d a positive integer with d|n). Here mu denotes the Mobius mu function. I don't know much about {b_n}'s properties in relation to {a_n}, but it may be a place to look. Note that b_n does not need to be an integer, since we could have a_1=1, a_2=8, a_3=12, a_6=24 (and some extension to the rest of the integers) and b_6=1/4.

---- David