For a given integer $n$, I am interested in the number of different numerical semigroups one can make with a generating set consisting only of integers in $[n]$.

I have done some small examples. For $n=1$ there is only $\langle 1\rangle$ and the same goes for $n=2$. For $n=3$ there are $\langle 1\rangle$, $\langle 2,3\rangle$, and $\langle 3,4\rangle$. For $n=4$ I won't list all the ones I found but I believe there to be $7$ of them, and for $n=5$ I found $8$ (of course I may have made a mistake).

I have found references online counting numerical subgroups by multiplicity $m$ and genus $g$, but was not able to find anything on this variant of the counting problem. In fact, it would be really great if I could figure out how to count the number of numerical semigroups with generating set in $[n]$ and with genus $g$, for given integers $n$ and $g$. Any remarks or pointers to the literature would be greatly appreciated. Thanks in advance!

For a given integer $n$, I am interested in the number of different numerical semigroups one can make with a generating set consisting only of integers in $[n]$.

I have done some small examples. For $n=1$ there is only $\langle 1\rangle$ and the same goes for $n=2$. For $n=3$ there are only $\langle 1\rangle$ and $\langle 2,3\rangle$ and for $n=4$ there are $\langle 1\rangle$, $\langle 2,3\rangle$, and $\langle 3,4\rangle$. For $n=5$ I won't list all the ones I found but I believe there to be $7$ of them, and for $n=6$ I found $8$ (of course I may have made a mistake).

I have found references online counting numerical subgroups by multiplicity $m$ and genus $g$, but was not able to find anything on this variant of the counting problem. In fact, it would be really great if I could figure out how to count the number of numerical semigroups with generating set in $[n]$ and with genus $g$, for given integers $n$ and $g$. Any remarks or pointers to the literature would be greatly appreciated. Thanks in advance!