 Solution

The allowed primes are exactly $2,3,5,7$. DefineThe fundamental theorem of arithmetic makes this definition unambiguous and proves injectivity. For every $(a,b,c,d)\in {\mathbb{N}}^4$, the positive integer $2^a{3}^b{5}^c{7}^d$ belongs to $S$ and maps to that tuple, proving surjectivity. The inverse function is explicitly ${\Phi }^{-1}(a,b,c,d)=2^a{3}^b{5}^c{7}^d$. In particular, $1$ maps to $(0,0,0,0)$. This is the prime-exponent parametrization of smooth numbers.