smc

The first few terms are $a_1,a_2,a_1{a}_2,a_1{a}_2^2,a_1^2{a}_2^3,\dots$ . If this is a geometric progression, $\frac{a_2}{a_1}=a_1=a_2=a_1{a}_2$. $a_1=0,1$, $a_2=0,1$. Since this is a sequence of positive integers, then the answer must be $\boxed{\text{if and only if }{a}_1=a_2=1}$.