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Source. Theorem 3, p. 15, of Jean-Marie De Koninck, Florian Luca and Igor E. Shparlinski, Powerful numbers in short intervals, Bull. Austral. Math. Soc. 71 (2005), 11--16, doi:10.1017/S0004972700037953. See the source card.

Read depth. Claims checked: the statement was read clause by clause on the printed page. The proof (p. 15) was read for structure only. Nothing here is independently reviewed.

Statement

Theorem 3 (p. 15). For any positive integers $L$ and $K$, the interval $(L,L+K)$ contains at most

$$O\left(\frac{K\log\log K}{\log K}\right)$$

squarefull numbers.

The implied constant is absolute. Since a number that is $\kappa$-full for some $\kappa\ge2$ is squarefull, the same bound counts the integers in $(L,L+K)$ that are $\kappa$-full for at least one $\kappa\ge2$, which is how the introduction (p. 12) presents it. The statement is printed for all positive $K$; the bound is meaningful once $\log\log K>0$.

Proof pointer

P. 15. With $w=\log K/\log\log K$, the squarefull numbers in the interval are split by whether they have a prime factor $p$ with $w\le p\le K$. Those that do are divisible by $p^2$ and number $\ll K/\log K$; those that do not are counted by the Brun sieve, giving $\ll K\prod_{w\le p\le K}(1-1/p)\ll K\log\log K/\log K$.

Dependencies

The Brun sieve, cited as H. Halberstam and H.-E. Richert, Sieve methods (Academic Press, 1974), Theorem 2.2.

Bears on

• Problem 942: taking $L=n^2$ and $K=2n+1$ gives at most $1+O(n\log\log n/\log n)$ powerful integers in $[n^2,(n+1)^2)$ for every $n\ge1$. This upper bound is a power of $n$, far above the bound $(\log n)^{c+o(1)}$ the problem asks about, so it does not settle the problem. The paper does not mention the problem.