# [seqfan] Range of one-to-one multiplicative functions

franktaw at netscape.net franktaw at netscape.net
Wed Jun 10 00:43:52 CEST 2009

```I've been looking a bit recently at what sets can be the set of values
taken on by one-to-one multiplicative functions.  For example, it is
easy to find multiplicative functions onto the positive integers; one
is induced by any permutation of the primes, or by any permutation of
the positive integers (acting on the exponents), or by any permutation
of A050376.

Of course, the kth powers for any k are obtainable, as a(n) = n^k is
multiplicative.  Less obviously, the powers of k can be obtained as
k^A052331(n).  We can obtain the powerful numbers with A064549(n), and
the square-free numbers with (my recently submitted) A160102(n).

(Note that without the constraint that the function be one-to-one, we
can map onto any set containing 1: just take a(2^k) = s(k), and a(p^k)
= 1 for any odd prime p.)

I have not been able to settle the question of whether there is a
one-to-one multiplicative function onto the non-primes (i.e., the
composite numbers union {1}).  I suspect that the answer is no, but I
haven't been able to prove it.  A proof (or counter-example) would be
appreciated.