Consecutive Integers with Equally Many Principal Divisors
نویسندگان
چکیده
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 of 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+1 is not a divisor of n). The standard notation pα||n can be read as “pα is a principal divisor of n”. For each integer n ≥ 0, let Pn be the set of all positive integers with exactly n principal divisors, so P0 = {1}, and P1 = {2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, · · · }, P2 = {6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, · · · }, P3 = {30, 42, 60, 66, 70, 78, 90, 102, 105, 110, · · · }, P4 = {210, 330, 390, 420, 462, 510, · · · }, etc. In particular, P1 comprises the prime-powers, or principal integers; P2 comprises the products of two coprime principal integers, or rank 2 integers; and so on. Collectively, we call {Pn : n ≥ 0} the rank sets of positive integers.
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