Problem 1081

Let $A(x)$ count the number of $n\leq x$ which are the sum of two squarefull numbers (a number $m$ is squarefull if $p\mid m$ implies $p^2\mid m$ ). Is it true that

A(x) \sim c \frac{x}{\sqrt{\log x}}

for some $c>0$ ?

Source: erdosproblems.com/1081

Number theory
Powerful numbers

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SolvedDisproved

An accepted solution exists. The statement is false.

BaBr94: Baker, R. C. and Brüdern, J., On sums of two squarefull numbers. Math. Proc. Cambridge Philos. Soc. 116 (1994), no. 1, 1-5.
Bl04: Blomer, Valentin, Binary quadratic forms with large discriminants and sums of two squareful numbers. J. Reine Angew. Math. 569 (2004), 213-234.
BlGr06: Blomer, Valentin and Granville, Andrew, Estimates for representation numbers of quadratic forms. Duke Math. J. 135 (2006), no. 2, 261-302.
Er76e: Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen (1976), 271-282.
Od81: Odoni, R. W. K., A problem of Erdős on sums of two squarefull numbers. Acta Arith. 39 (1981), no. 2, 145-162.

Problem 1081

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