Consecutive Integers with Equally Many Principal Divisors

Abstract

Classifying the positive integers as primes, composites, and the unit, is so familiar that it seems inevitable. However, other classifications can bring interesting relationships to our attention. In that spirit, let us classify positive integers by the number o? principal divisors they possess, where we define a principal divisor of a positive integer n to be any prime-power divisor pa n which is maximal (so p is prime, a is a positive integer, and pa+l is not a divisor of n). The standard notation pa \\n can be read as /?fl is a principal divisor of n. 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. (Recall that a multiset is a collection of elements in which multiple occurrences are permitted.) 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. Consequently every positive integer is the product of its principal divisors, and every finite set of powers of distinct primes is the set of principal divisors of a unique positive integer. Of course, the number of principal divisors of n is equal to the number of distinct prime factors of n, but here the principal divisors are the simple structural components of n, whereas the distinct prime factors are but a shadow of that structure. Readers who find the present paper of interest might find similar interest in [6], where upper bounds on the sum of principal divisors of n are established by elementary means. For each integer n > 0, let Pn be the set of all positive integers with exactly n principal divisors, so Pq = {1}, and