On Sat, 3 May 2003, David Wilson wrote:
This question fell out of an investigation as to whether there exist two consecutive integers of the form p^2 q^3 with p, q distinct primes.
David, did this come from the following more general problem? Let E=[e_1,e_2,...,e_k] be a list of integers satisfying 1 <= e_1 <= e_2 <= . . . <= e_k. Let S(E) be the set of all positive integers n such that n = Prod{p_i^e_i} where the p_i's are distinct primes. Say that the integers in S(E) have factorization pattern E. I believe this is a known concept. Which of the sets S(E) contain at least two consecutive integers? Your question is for E = [2,3]. It's easy to find E's which contain 1's for with S(E) contains two consecutive integers, but a search of n <= 2 million shows no such S(E)'s when each e_i > 1. In fact if one checks the minimum gap, n-m, with n,m in S(E), the smallest gap in this range is 5 and it occurs for your case E=[2,3]. Based on this scant evidence maybe [2,3] is the most promising? Is it the case that if 1 is not in E then S(E) contains no consecutive pair of integers? To generalize further: Let S be any set of integers and let f be any integer valued function. Ask for the number of n in S such that f(n) is also in S. The Catalan problem was for S = the set of non-trivial powers and f(n) = n + 1. Maybe this is all old hat? In the above one might replace n+1 by n+2 or n+3 or ...? --Edwin