Difference of Two Cubic Numbers

I'm surprised that I've not dealt with this topic before but as far as I can tell I haven't. The topic in question is numbers that are the difference of two cubes, or more specifically the difference of two positive cubes. My diurnal age today is 27872, a palindrome, with the property that:$$27872 = 38^3-30^3$$It's easy enough to develop an algorithm to determine all such numbers in the range up to 40000 and the total is 825. However, if we consider only those numbers equal to or greater than 27872, then only 188 numbers qualify. They are (permalink):

27872, 27937, 28063, 28415, 28460, 28519, 28568, 28656, 28672, 28701, 28737, 28791, 28828, 28854, 29051, 29062, 29078, 29080, 29107, 29279, 29393, 29402, 29448, 29528, 29575, 29617, 29666, 29701, 29727, 29735, 29763, 29764, 29783, 29790, 30016, 30024, 30043, 30105, 30248, 30301, 30312, 30483, 30571, 30708, 30807, 30907, 30970, 31024, 31031, 31040, 31085, 31106, 31213, 31228, 31232, 31304, 31437, 31519, 31768, 31806, 31841, 31869, 31976, 32039, 32137, 32227, 32256, 32319, 32425, 32445, 32464, 32465, 32552, 32562, 32643, 32704, 32741, 32760, 32761, 32767, 32832, 32851, 32858, 32920, 32949, 32984, 33077, 33193, 33336, 33391, 33472, 33614, 33724, 33740, 33752, 33875, 34027, 34047, 34209, 34391, 34489, 34531, 34606, 34658, 34669, 34784, 34875, 34902, 34930, 34937, 35008, 35028, 35163, 35189, 35208, 35315, 35317, 35425, 35576, 35594, 35721, 35812, 35873, 35910, 35929, 35936, 35971, 36008, 36016, 36153, 36253, 36297, 36316, 36504, 36506, 36560, 36631, 36632, 36785, 36829, 37000, 37043, 37107, 37296, 37297, 37367, 37395, 37448, 37449, 37576, 37648, 37962, 37969, 37973, 38017, 38142, 38151, 38285, 38304, 38402, 38486, 38528, 38575, 38619, 38647, 38656, 38779, 38792, 38961, 39004, 39088, 39130, 39179, 39240, 39247, 39277, 39296, 39303, 39331, 39368, 39500, 39611, 39636, 39797, 39807, 39815, 39816, 39823 (see OEIS A181123)

Of these 188, there are five numbers that can be expressed as a difference of two cubes in more that one way. These are 27937, 28063, 34209, 35208 and 35929. The details are as follows:$$ \begin{align} 27937 &= 33^3- 20^3 \\ &=97^3- 96^3 \\ 28063 &= 31^3- 12^3 \\ &=40^3- 33^3 \\34209 &= 33^3- 12^3 \\ &=40^3- 31^3 \\ 35208 &= 33^3- 9^3 \\ &= 34^3- 16^3 \\ 35929 &= 33^3- 2^3 \\ &=34^3- 15^3 \end{align}$$Notice that 27937 is a difference of successive cubes but, because it is not prime, it cannot be a Cuban prime. The only Cuban prime in the range is 33391where:$$33391=106^3-105^3$$I've dealt with this category of primes in my blog post titled Cuban Primes way back in July of 2016.