A cube-full number is a positive integer 
such that 
implies 
for every prime number 
. Thus cube-full numbers are a subclass
of the powerful numbers. More generally, a positive integer is 
-full if 
implies 
.
The first few cube-full numbers are 1, 8, 16, 27, 32, 64, 81, 125, 128, 216, 243, 256, 343, 432, 512, 625, 648, 729, 864, 1000, ... (OEIS A036966).
Let 
be the set of positive cube-full numbers. Beyer de Ryke (2026)
proved that, for every 
,
there is a residue class relatively
prime to its modulus in which 
has upper relative density at most 
. Consequently, the positive
integers that are not sums of at most three cube-full numbers have positive lower
natural density. This proves the infinitude assertion
in Erdős problem 940 for three cube-full summands,
while the stronger question of whether the representable integers
have natural density zero remains open.
