On the determinants of divisor matrices

The family {An}n∈N of divisor matrices was introduced by Raphael Yuster (Discrete Math. 224 (2000) 225–237) as a tool in the investigation of magic sequences. He conjectured that the An are non-singular over any field (so det An=±1) and that det An=(−1)n−1, and proved the first conjecture for n divisible by at most two primes. It was proved for arbitrary n by this author (Discrete Math. 250 (2002) 125–135). We here prove that the second conjecture holds for n with at most two prime divisors and for even n with three prime divisors. We also prove that it holds for n if it holds for all square-free divisors of n.