PUMaC 2022 · 数论（B 组） · 第 2 题

(2) (3) (4) ∞ ∞ ∞ sequences { a } , { a } , and { a } . Find the sum of the elements of S that lie in n n n n =1 n =1 n =1 the interval [1 , 100]. Proposed by Sunay Joshi Answer: 4451 ( ℓ ) 1 ∞ ℓ We claim that a number k + 1 is skipped by the sequence { a } iff k + 1 = m + ⌈ ( m + ) ⌉ n n =1 2 for some m ≥ 0. To see this, suppose k + 1 is skipped by the sequence, so that a = k and n √ 1 ℓ a ≥ k + 2. The condition a = k is equivalent to k ≤ n + n + < k + 1 and thus n +1 n 2 1 1 k ℓ ( m − ) ≤ n < ( m + ) , where m = k − n . The condition a ≥ k + 2 is equivalent n +1 2 2 √ ℓ 1 1 ℓ to k + 2 ≤ ( n + 1) + n + 1 + , which can be rewritten as ( m + ) − 1 ≤ n . Combining 2 2 1 ℓ these two inequality chains, we find that n = ⌈ ( m + ) ⌉ − 1, hence the skipped number is 2 1 ℓ k + 1 = m + ⌈ ( m + ) ⌉ , as claimed. 2 1 2 2 It follows that the numbers skipped in the sequence for ℓ = 2 are m + ⌈ m + m + ⌉ = ( m + 1) ; 4 3 3 1 3 3 1 3 2 3 2 the numbers skipped for ℓ = 3 are m + ⌈ m + m + m + ⌉ = m + m + ⌈ m + m + ⌉ ; 2 4 8 2 4 8 4 3 3 2 1 1 4 and the numbers skipped for ℓ = 4 are m + ⌈ m + 2 m + m + m + ⌉ = m + m + 2 2 16 3 3 2 1 1 2 m + ⌈ m + m + ⌉ . The skipped numbers for ℓ = 2 are 1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100, 2 2 16 the skipped numbers for ℓ = 3 are 1 , 5 , 18 , 46 , 96, and the skipped numbers for ℓ = 4 are 1 , 7 , 42. The sum of (distinct) numbers that are skipped in at least one of the sequences can be seen to be 599, hence the sum of the numbers in [1 , 100] that are not skipped in any list is 5050 − 599 = 4451, our answer. 3