Problem Statement
Let $r\geq 3$. A number $n$ is $r$-powerful if for every prime $p$ which divides $n$ we have $p^r\mid n$.
Are there infinitely many integers which are not the sum of at most $r$ many $r$-powerful numbers? Does the set of integers which are the sum of at most $r$ $r$-powerful numbers have density $0$?
Are there infinitely many integers which are not the sum of at most $r$ many $r$-powerful numbers? Does the set of integers which are the sum of at most $r$ $r$-powerful numbers have density $0$?
Categories:
Number Theory Powerful
Progress
Erdős [Er76d] claims that the claim that the set has density $0$ is 'easy' for $r=2$ (a potential 'easy argument' is given in the comments by Tao; this was first proved in the literature by Baker and Brüdern [BaBr94]). For $r=3$ it is not even known if those integers which are the sum of at most three cubes has density $0$.In the Oberwolfach problem book this is recorded in 1986 as a problem of Erdős and Ivić.
In [Er76d] Erdős claims that 'a simple counting argument' implies that there are infinitely many integers which are not the sum of at most $r$ many $r$-powerful numbers, but Schinzel pointed out he made a mistake.
Heath-Brown [He88] has proved that all large numbers are the sum of at most three $2$-powerful numbers, see [941].
See also [1081] for a more refined question concerning the case $r=2$, and [1107] for the case of $r+1$ summands.
Source: erdosproblems.com/940 | Last verified: January 19, 2026