For a natural number n, let $1 = d_1 < \dotsc < d_{\tau(n)} = n$ denote the divisors of n in increasing order. Does it hold that $\sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i} \ll 1 + \sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i}$ for $n \to \infty`$, i.e. Unparsed notation: \sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i} \in O \left( 1 + \sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i}) Unparsed notation: \right)?

For a natural number n, let $1 = d_1 < \dotsc < d_{\tau(n)} = n$ denote the divisors of n in increasing order. Does it hold that $\sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i} \ll 1 + \sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i}$ for $n \to \infty`$, i.e. Unparsed notation: \sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i} \in O \left( 1 + \sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i}) Unparsed notation: \right)?

This conjecture has been disproved: - In September 2025, Terence Tao gave a conditional _negative_ answer assuming the prime tuples conjecture, see erdos_884_false_of_hardy_littlewood for this implication. - Daniel Larsen subsequently gave an unconditional disproof.

*Reference:* erdosproblems.com/884