PUMaC 2017 · 数论（B 组） · 第 4 题

PUMaC 2017 — Number Theory (Division B) — Problem 4

The sequence of positive integers a , a , . . . has the property that gcd( a , a ) > 1 if and only 1 2 m n if | m − n | = 1. Find the sum of the four smallest possible values of a . 2

解析

Assume all exponents considered are at least 1. For n > 1, a must have at least two distinct n prime factors (so that a has one prime in common and a has another); it is clear that n − 1 n +1 a b we can construct a sequence satisfying those requirements given a = cp q for p, q prime but 2 c not necessarily prime, by choosing no other element in the sequence to share prime factors with c and by letting p | a and q | a . We now simply wish to find the four smallest numbers 1 3 a b of the form cp q . Looking at numbers of the form pq : 6 = 2 · 3 , 10 = 2 · 5 , 14 = 2 · 7 , 15 = 3 · 5. 2 However, 12 = 2 · 3 is also of the desired form, but any other value of c gets a > 15 (e.g., 2 pq = 6, c = 3 = ⇒ a = 18. So, our desired value is 6 + 10 + 12 + 14 = 42 . 2 Problem written by Zack Stier