Prime signature functions

[David Wilson]

I notice that for some sequences, a(n) is a function of the prime signature 
of a n, that is, a function of the multiset of exponents in the prime 
factorization of n.

For such a function, a(p) will be the same for all primes p, a(pq) will be 
the same for any two distinct primes p, a(p^2) will be the same for any 
prime p, etc.

For the moment I will call there prime signature functions.  Is there a 
standard name for such functions?

There are prime signature functions which are not multiplicative and vice 
versa.  Examples of prime signature functions that are not multiplicative 
are A001221 (number of prime divisors of n) and A001222 (number of prime 
divisors with multiplicity).  Examples of multiplicative functions that are 
not prime signature functions are A000010 (totient) and A000203 (sum of 
divisors).  A function that is both prime signature and multiplicative is 
A000005 (number of divisors of n).  A function is both prime signature and 
mutliplicative if f(p^e) is strictly a function of e.

What made me think of this is that I was recently working on A000028. 
Membership of n in A000028 depends entirely on the prime signature of n, so 
its membership function is a prime signature function.

- David W. Wilson

"Truth is just truth -- You can't have opinions about the truth."
   - Peter Schickele, from P.D.Q. Bach's oratorio "The Seasonings"