Claims

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1981_01_01_odoni: Odoni shows that the count of integers up to x that are sums of two squarefull numbers exceeds any constant times x over the square root of log x, so Problem 1081's asymptotic fails; refereed and credited by the site.

2004_01_30_blomer: Blomer (J. Reine Angew. Math., 2004) bounds the count of sums of two squarefull numbers up to x below by x over log x to the power 0.253, which exceeds any constant times x over the square root of log x; refereed.

2006_11_01_blomer_granville: Blomer and Granville (Duke Math. J., 2006) bound the count of sums of two powerful numbers up to x within powers of log log x of x over log x to the power 1 - 2^(-1/3), so the square-root-of-log asymptotic fails; refereed.

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