I'm working on a project, but i'm stuck because i would need to count the different partitions of an integer which verify a certain property. I've never seen anyone looking at such a kind of partitions before.
Lets note $\pi(n)$ a partition of $n$ i.e. $$\sum_{n\in \pi(n)}=n$$ which verify the following properties :
$1 \in \pi(n)$
$4\notin \pi(n)$
all elements of $\pi(n)$ are different from each others
$\forall q\in \pi(n) \quad \forall d\mid q \quad d\in\pi(n)$.
For example $\pi(12)=(1,2,3,6)$ is a suitable partition, but $(1,4,7)$ or $(1,3,8)$ aren't.
Is there a way do count this kind or partitions ?
Could we find how much there is when $n$ tends to infinity ?