An n-divisor arithmetic progression of sub-square-root logarithmic height must have length at least roughly n^{3/4}/sqrt(log n), which disproves the arithmetic-progression version of the Strong Divisor Conjecture at (1/3,1/3).
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Refuting a Conjecture of Umans and Wang on Arithmetic-Progression Divisor Covers
An n-divisor arithmetic progression of sub-square-root logarithmic height must have length at least roughly n^{3/4}/sqrt(log n), which disproves the arithmetic-progression version of the Strong Divisor Conjecture at (1/3,1/3).