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"