We start with a little-known Euler type theorem (due to Alladi) which is the following:

The number of partitions of n into distinct parts not divisible by k (i.e. k-regular partitions with distinct parts) equals the number of partitions of n into odd parts none repeated more than k-1 times.

k=1 and 2 are tautologies.  k=3 plays a prominent role in Schur’s 1926 partition theorem.  Both Alladi and Schur have further partition identities related to k=2 which we will discuss.  Obviously, k = infinity is Euler’s theorem.  We then proceed to k=4 where an empirical investigation leads to a result for overpartitions.  We conclude with a proof of the k=4 case and look at results and possibilities for k>4.