HMMT 十一月 2019 · 冲刺赛 · 第 23 题

HMMT November 2019 — Guts Round — Problem 23

[12] For a positive integer n , let, τ ( n ) be the number of positive integer divisors of n . How many integers 1 ≤ n ≤ 50 are there such that τ ( τ ( n )) is odd? ◦ ◦

解析

[12] For a positive integer n , let, τ ( n ) be the number of positive integer divisors of n . How many integers 1 ≤ n ≤ 50 are there such that τ ( τ ( n )) is odd? Proposed by: Kevin Liu Answer: 17 Note that τ ( n ) is odd if and only if n is a perfect square. Thus, it suffices to find the number of integers 2 n in the given range such that τ ( n ) = k for some positive integer k . If k = 1, then we obtain n = 1 as our only solution. If k = 2, we see that n is either in the form pq 3 or p , where p and q are distinct primes. The first subcase gives 8 + 4 + 1 = 13 solutions, while the second subcase gives 2 solutions. k = 3 implies that n is a perfect square, and it is easy to see that 2 only 6 = 36 works. Finally, k ≥ 4 implies that n is greater than 50, so we’ve exhausted all possible cases. Our final answer is 1 + 13 + 2 + 1 = 17. ◦ ◦