Dense sets of natural numbers with unusually large least common multiples

We construct a set A ⊂ N such that one has ∑n ∈ A: n ≤ x 1/n = exp((1/2+o(1)) (loglog x)1/2 logloglog x ) and 1(∑n ∈ A: n ≤ x 1/n)² ∑n,m ∈ A: n < m ≤ x 1/lcm(n,m) 1 for sufficiently large x. The exponent 1/2 can replaced by any other positive constant, but the growth rate is otherwise optimal. This answers in the negative a question of Erd{o}s and Graham.