In recent work with Raum, the authors considered congruences for the ordinary partition function p(n) of the form \(p(\ell Q^r n+\beta )\equiv 0\pmod \ell \) where \(\ell , Q\ge 5\) are prime and \(r\in \{1,2\}\) , and proved a number of results which show that such congruences are scarce in a precise sense. Here we improve one of our results when \(r=1\) ; in particular, we prove (outside of trivial cases) that the set of primes Q such that there exists \(\beta \in \mathbb {Z}\) with \(p(\ell Q n+\beta )\equiv 0\pmod \ell \) for all n has density zero. The proof involves a modification of part of our previous argument and an application of a recent theorem of Dicks regarding modular forms of half-integral weight and level one modulo \(\ell \) .
