On the number of multiplicative sub-monoids of integers mod $n$

Question

Let $n \geq 2$ be given. Is there a formula describing the number of multiplicative closed subsets of the ring $\mathbb{Z}_n$ ?

Answer 1

This is a surprisingly difficult problem. In $\mathbb{Z}_n$, the number of subsets closed under multiplication is given by sequence A272213 of the OEIS. Starting from $n=1$, it goes like this:

1, 3, 5, 8, 7, 26, 9, 29, 20, 42, 9, 168, 13, 58, 106, 141, 11, 220, 13, 504, 141, 58, 9, 1997, 66, 99, 279, 1420, 13, 2764, 17, 2171, 141, 83, 248, 12412, 19, 99, 258, 19549, 17, 4135, 17, 15508, 8310, 58, 9, 91508, 228, 1758, 226, 60303, 13, 26691, 248, 228792, 245, 99, 9, 890397, 25, 140, 109296, 389930

The OEIS mentions that if $p$ is prime, then the number for $\mathbb{Z}_p$ is $2\tau(p-1)+1$, where $\tau(m)$ is the number of divisors of $m$. This is easily deduced from the fact that $\mathbb{Z}_p^*$ is a cyclic group. For non-prime numbers, there seems to be no known formula.

It is worth mentioning (as the OEIS does) that the following paper contains a lot of information on the sub-semigroups of the multiplicative semigroup $\mathbb{Z}_m$: Edwin Hewitt and Herbert S. Zuckerman, The multiplicative semigroup of integers modulo $m$ , Pacific J. Math., Vol. 10, 1960.