A recurrence relation;lcm(lowest common multiple) links to the n! Factorial.
Let $s(1)=1$
$s(n)=s(n-1)+lcm[n,s(n-1)]+n$, for $n\ge2$
$s(n)=(n+1)!-1$
I try:
I have no idea where to start to tackle this problem, but at least show that it is work.
$s(2)=1+lcm[2,1]+2=5$, $s(2)=3!-1=5$
$s(3)=5+lcm[3,5]+3=23$, $s(3)=4!-1=23$
$s(4)=23+lcm[4,23]+4=119$, $s(4)=5!-1=119$
$s(5)=119+lcm[5,119]+5=719$, $s(5)=6!-1=719$
And so on...