Andrews [Generalized Frobenius Partitions, Memoirs of the American Mathematical Society, 301 (American Mathematical Society, Providence, RI, 1984)] defined two families of functions,
$\phi _k(n)$
and
$c\phi _k(n),$
enumerating two types of combinatorial objects which he called generalised Frobenius partitions. Andrews proved a number of Ramanujan-like congruences satisfied by specific functions within these two families. Numerous other authors proved similar results for these functions, often with a view towards a specific choice of the parameter
$k.$
Our goal is to identify an infinite family of values of k such that
$\phi _k(n)$
is even for all n in a specific arithmetic progression; in particular, we prove that, for all positive integers
$\ell ,$
all primes
$p\geq 5$
and all values
$r, 0 < r < p,$
such that
$24r+1$
is a quadratic nonresidue modulo
$p,$
$$ \begin{align*} \phi_{p\ell-1}(pn+r) \equiv 0 \pmod{2} \end{align*} $$
for all
$n\geq 0.$
Our proof of this result is truly elementary, relying on a lemma from Andrews’ memoir, classical q-series results and elementary generating function manipulations. Such a result, which holds for infinitely many values of
$k,$
is rare in the study of arithmetic properties satisfied by generalised Frobenius partitions, primarily because of the unwieldy nature of the generating functions in question.