"logarithmetic" sequences

[Marc LeBrun]

Speaking of multiplicative sequences, ie those that (perhaps with 
conditions) obey

   a(mn) = a(m) a(n)

I was wondering what might be said for sequences that instead satisfy

   a(mn) = a(m) + a(n)

which might be called "log-arithmetic" (or "logarithm-etic" if you prefer) 
by analogy with regular logs.

Have these been discussed here before?  Is there any standard name for them?

Of course, as with their multiplicative cousins, logarithmetic sequences 
can be generated by their prime-power index subsequence.

So by summing f(p^e) for various easy f we get A001221, A001222, A056169, 
A001414, and A008472 for instance.

Generalizing + to g(.) we can further posit "factorization-defined" sequences

   a(n) = g(f(p1^e1),f(p2^e2),...)

for various f & g, and even imagine superseekeresque detection of 
them.  (And even more generally, to use other unique expansions).

More practically, if someone has the wherewithal to crunch the database, it 
might be interesting to simply collect all the existing instances of 
candidate logarithmetic sequences, since there seem to be quite a few.

It might be even more immediately useful to check for any entries 
potentially missing their "mult" keywords.

(Just to guiltily throw a few more suggestions onto the endless pile!)