We consider the ``visible'' Wigner matrix, a Wigner matrix whose $(i, j)$-th entry is coerced to zero if $i, j$ are co-prime. Using a recent result from elementary number theory on co-primality patterns in integers, we show that the limiting spectral distribution of this matrix exists, and give explicit descriptions of its moments in terms of infinite products over primes $p$ of certain polynomials evaluated at $1/p$. We also consider the complementary ``invisible'' Wigner matrix.