Consecutive integers with equally many principal divisors

The Fundamental Theorem of Arithmetic is usually stated in a form emphasizing how primes enter the structure of the positive integers, such as: Every positive integer is the product of a unique finite multiset of primes. Alternatively, the Fundamental Theorem of Arithmetic can be stated in a form that focuses on how maximal prime-powers enter the structure of the positive integers, such as: Every positive integer is the product of a unique finite set of powers of distinct primes. Here, Eggleton and MacDougall classify positive integers by the number of principal divisors they possess, where they define a principal divisor of a positive integer n to be any prime-power divisor psupa |n which is maximal. They found that for every n greater than or less than 1 there are only finitely many runs of size greater than N in Psubn, where N is the product of the first n primes.