PARTITIONS INTO PRIME POWERS

For a subset A subset of N, let pA(n) denote the restricted partition function which counts partitions of n with all parts lying in A. In this paper, we use a variation of the Hardy-Littlewood circle method to provide an asymptotic formula for pA(n), where A is the set of kth powers of primes (for fixed k). This combines Vaughan's work on partitions into primes with the author's previous result about partitions into kth powers. This new asymptotic formula is an extension of a pattern indicated by several results about restricted partition functions over the past few years. Comparing these results side-by-side, we discuss a general strategy by which one could analyze pA(n) for a given set A.