A term $a_n$ can only be equal to $1$ for $n=4$, and that index is excluded. We check by hand the values for $n<5$, so assume $n\geq 5$. Say $a_n$ is composite, and let $p$ be the least prime dividing it. Then $p^2 \leq a_n = n(n+1)-19 < (n+1)^2$, so $p\leq n$. Let $0\leq k < n$ be the residue of $n$ modulo $p$; then $a_k \equiv a_n \equiv 0 \pmod{p}$, and so $p\mid \gcd (a_n,a_k)$, contradiction. As we can see, the value $19$ plays no role; its only merit is that it produces quite many primes! There was an even better such number, known to Euler, namely $-41$.