Multiply or Divide by n
Jelly, 6 bytes
R×:&lt;?/

A monadic Link accepting a positive integer, \$n\$, which yields a positive integer, \$a(n)\$.
Try it online! Or see the test-suite.
How?
R×:&lt;?/ - Link:
R      - range -&gt; [1..n]
     / - reduce by (i.e. evaluate f(f(...f(f(f(1,2),3),4),...),n) with this f(a,b):
    ?  -   if...
   &lt;   -   ...condition: (a) less than (b)?
 ×     -   ...then: multiply -&gt; a×b
  :    -   ...else: integer divide -&gt; a//b


Output the sequence up to \$a(n)\$ with:
R×:&lt;?\