Strong divisibility and lcm-sequences

Let $R$ be a gcd-domain (for example let $R$ be a unique factorization domain), and let $(a_n{)}_{n⩾1}$ be a sequence of nonzero elements in $R$. We prove that $\gcd (a_n,a_m)=a_{\gcd (n,m)}$ for all $n,m⩾1$ if and only if $$a_n=\prod _{d\mid n}{c}_d\text{quadfor}n⩾1,$$ where $c_1=a_1$ and $c_n=lcm(a_1,a_2,\dots ,a_n)\mathrm{/}lcm(a_1,a_2,\dots ,a_{n-1})$ for $n⩾2$. All equalities with gcd and lcm are determined up to units of $R$.