If we introduce k in the name "(sum of all previous terms)/k", then cases k=1,2 correspond to
A070218,
A196375, and in general case, the sequence begins with k 2's, with gradually (not monotonically) decreasing multiplicity of terms; e.g., at case k=10 the sequence begins: 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 11, 11, 11, 11, 13, 17, 17, 17, 19, 23, 23, 29, 29, 31, 37, 41.
