Starting from the identity $r=(x-d)/2$, relating an odd integer $x$ and a divisor $d$ of $x^2$ through the inradius of an integer right triangle, we introduce a modular sieve acting on the index $r$. Divisibility conditions on $x=2r+1$ are translated into explicit congruence exclusions $r\equiv (d-1)/2 \pmod d$, yielding a recursive modular structure equivalent to a wheel sieve. The surviving indices generate the odd primes in increasing order via $p=2r+1$, and the modular formulation admits efficient residue-based implementations. This feature makes the method particularly suitable for accessing primes at large indices with a minimal memory footprint, without requiring storage of all preceding primes. The focus is on structural and modular properties; computational considerations are included only insofar as they follow from the modular formulation.