Multiples and Divisors

Before discussing multiplication, let us speak about addition. The number A(k) of distinct sums i+ j ≤ k such that 1 ≤ i ≤ k/2, 1 ≤ j ≤ k/2 is clearly 2 bk/2c − 1. Hence the number A(2n) of distinct elements in the n × n addition table involving {1, 2, . . . , n} satisfies limn→∞A(2n)/n = 2, as expected. We turn to multiplication. Let M(k) be the number of distinct products ij ≤ k such that 1 ≤ i ≤ √ k, 1 ≤ j ≤ √ k. One might expect that the numberM(n) of distinct elements in the n×nmultiplication table to be approximately n/2; for example, M(10) = 42. In a surprising result, Erdös [1, 2, 3] proved that limn→∞M(n)/n = 0. More precisely, we have [4]