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Claim. Let $V(x)=A(x)$ count the integers $n\le x$ that are sums of two squarefull (powerful) numbers. Corollary 2 of V. Blomer and A. Granville, Estimates for representation numbers of quadratic forms, Duke Math. J. 135 (2006), no. 2, 261–302, states that for some $A\in\mathbb R$

$$\frac{x(\log\log x)^A}{(\log x)^{1-2^{-1/3}}}\ll V(x)\ll \frac{x(\log\log x)^{2^{2/3}-1}}{(\log x)^{1-2^{-1/3}}},$$

as recorded on the source card blomer_2006_estimates_representation_numbers_quadratic_forms. Since $1-2^{-1/3}\approx0.2063<1/2$, the lower bound alone gives $A(x)\sqrt{\log x}/x\to\infty$, so $A(x)$ is not asymptotic to $c\,x/\sqrt{\log x}$ for any $c>0$. This disproves Problem 1081 independently of [[problems/diophantine_problems/E1081/claims/1981_01_01_odoni|Odoni's earlier disproof]], and it fixes the order of $A(x)$ up to powers of $\log\log x$.

Depends on. No other wiki page; the claim rests on the cited paper.

Acceptance. Refereed: the paper appeared in the Duke Mathematical Journal, volume 135 (2006), published online on 1 November 2006 according to its Crossref record. Not reviewed: the site's curator credits Odoni with the disproof and cites this paper for the sharpest estimate of $A(x)$, which is not a credit for the disproof.