OEIS shows the number of groups of order $n$ upto $2047$. The Magma-online-calculator uses a database, but already for $1024,2004,2016,...$ it cannot determine the number of groups. Maple seems to calculate the number of groups (unless it is too large), but unfortunately, I do not have access to maple.
Does anyone know an online-calculator for the number-of-group-function, or a table with the known numbers upto $10,000$ or more ?
Or alternatively, allows PARI/GP, at least in principle, to calculate the number of groups of order $n$ ? I programmed the case $n$ squarefree, but I have no clue how to manage arbitary numbers $n$.
For the list of $n$'s which are included in the SmallGroups library, see : https://magma.maths.usyd.edu.au/magma/handbook/text/727
There are many missing n's which could be computed, even by hand (when $n=p^2q^2$, for example), but there are definitely some numbers less than $10000$ that are out of reach.
For example, the number of groups of order $2048$ is not actually known, higher powers of $2$ even less so, see for example https://www.math.auckland.ac.nz/~obrien/research/gnu.pdf
(This is a good reference for your question in general.)