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, for an integer 
, 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 problem 940 in the Erdős
problems collection for three cube-full summands, while the stronger question
of whether the representable integers have natural
density zero remains open.
