  Let $p(n)$ be the ordinary partition function. In the 1960s Atkin found a
number of examples of congruences of the form $p( Q^3 \ell
n+\beta)\equiv0\pmod\ell$ where $\ell$ and $Q$ are prime and $5\leq \ell\leq
31$; these lie in two natural families distinguished by the square class of
$1-24\beta\pmod\ell$. In recent decades much work has been done to understand
congruences of the form $p(Q^m\ell n+\beta)\equiv 0\pmod\ell$. It is now known
that there are many such congruences when $m\geq 4$, that such congruences are
scarce (if they exist at all) when $m=1, 2$, and that for $m=0$ such
congruences exist only when $\ell=5, 7, 11$. For congruences like Atkin's (when
$m=3$), more examples have been found for $5\leq \ell\leq 31$ but little else
seems to be known.
  Here we use the theory of modular Galois representations to prove that for
every prime $\ell\geq 5$, there are infinitely many congruences like Atkin's in
the first natural family which he discovered and that for at least $17/24$ of
the primes $\ell$ there are infinitely many congruences in the second family.
