Moufang loops of odd order $p_1p_2\dots p_nq^3$ with non-trivial nucleus

summary:It has been proven by F. Leong and the first author (J. Algebra {\bf 190} (1997), 474--486) that all Moufang loops of order $p^\alpha q_1^{\beta_1}q_2^{\beta_2}\cdot \cdot \cdot q_n^{\beta_n}$ where $p$ and $q_i$ are odd primes, are associative if $p<q_1<q_2<\cdot \cdot \cdot<q_n$, and \roster \item"(i)" $\alpha\leq 3$, $\beta_i\leq 2$; or \item"(ii)" $p\geq 5$, $\alpha\leq 4$, $\beta_i\leq2$. \endroster The first author also proved that if $p$ and $q$ are distinct odd primes, then all Moufang loops of order $pq^3$ are associative if and only if $q\not\equiv 1(\text{\rm mod}\, p)$ (J. Algebra {\bf 235} (2001), 66--93). In this paper, we prove that all Moufang loops of order $p_1p_2\cdot \cdot \cdot p_nq^3$ where $p_i$ and $q$ are odd primes, are associative if $p_1<p_2<\cdot \cdot \cdot <p_n