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Claim. Let $A(x)$ count the integers $n\le x$ that are sums of two squarefull numbers. There are constants $a,\beta,\gamma>0$ with

$$A(x) > a\,\frac{x}{\sqrt{\log x}} \exp\!\left(\beta\,\frac{\log\log x}{\log\log\log x}\right) \qquad (x>\gamma),$$

so $A(x)\sqrt{\log x}/x\to\infty$ and $A(x)$ is not asymptotic to $c\,x/\sqrt{\log x}$ for any $c>0$. This disproves Problem 1081. The result is Theorem 1 of R. W. K. Odoni, A problem of Erdős on sums of two squarefull numbers, Acta Arith. 39 (1981), 145–162; only its publication year is known, so the page is dated to the start of 1981. The site's remark prints Odoni's extra factor with the ratio inverted, as $\exp(c\log\log\log x/\log\log x)$, which tends to $1$ and would not refute the asymptotic; this page follows Theorem 1 of the paper. The source card is odoni_1981_problem_erdos_sums_two_squarefull_numbers.

Method. Every value $m^3x^2+n^3y^2$ with $x,y\geq1$ and $\gcd(m,n)=1$ is a sum of two squarefull numbers, so a lower bound for the count of integers these forms represent is a lower bound for $A(x)$; the reverse inclusion fails ($16=8+8$ is a value of no such form), and the argument does not need it. Odoni restricts to the forms with $m,n$ primes $p<q$ in a window depending on $x$, where the genus theory of discriminant $-4p^3q^3$ is tractable, proves a lower bound of order $x/\sqrt{\log x}$ for each such form uniformly (Theorem 2), bounds the pairwise overlaps (Theorem 3), and combines them by inclusion–exclusion.

Later refinements. Baker and Brüdern (Math. Proc. Cambridge Philos. Soc. 1994) proved the upper bound $A(x)\ll x(\log x)^{\varepsilon-1/6}$ for every $\varepsilon>0$. Blomer (J. Reine Angew. Math. 2004) proved $x(\log x)^{-0.253}\ll A(x)\ll x(\log x)^{-1/6}\log\log x$, and his sequel (J. London Math. Soc. 2005) proved $A(x)=x/(\log x)^{1-2^{-1/3}+o(1)}$. Blomer and Granville (Duke Math. J. 2006, Corollary 2 on the card blomer_2006_estimates_representation_numbers_quadratic_forms) pinned $A(x)$ to $x/(\log x)^{1-2^{-1/3}}$ up to powers of $\log\log x$, with $1-2^{-1/3}\approx0.2063$. The lower bounds of Blomer (2004) and of Blomer and Granville each refute the asymptotic on their own, and each has its own claim page: Blomer 2004 and Blomer and Granville 2006.

Acceptance. The paper is refereed: Acta Arith. 39 (1981). The site's curator, Thomas Bloom, marks Problem 1081 disproved and credits Odoni's paper for the disproof.

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