= Solution
The allowed <primes> are exactly $2,3,5,7$. Define
$$
\Phi(2^a3^b5^c7^d)=(a,b,c,d),\qquad a,b,c,d\in\mathbb N.
$$
The <fundamental theorem of arithmetic> makes this definition unambiguous and proves <injectivity>. For every $(a,b,c,d)\in\mathbb N^4$, the positive <integer> $2^a3^b5^c7^d$ belongs to $S$ and maps to that tuple, proving <surjectivity>. The <inverse function> is explicitly $\Phi^{-1}(a,b,c,d)=2^a3^b5^c7^d$. In particular, $1$ maps to $(0,0,0,0)$. This is the <prime-exponent parametrization of smooth numbers>.
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