1001
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13 = 1
23 = 8
33 = 27
43 = 64
53 = 125
63 = 216
73 = 343
83 = 512
93 = 729
103 = 1000
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13 cubes 4,9,16,25,36,46,64,81,100,121,144,169,196
Simply calculate the cubes of a few small numbers, and check which of them are in the desired range.
When it is of the form x3 + y3 or x3 - y3. x or y can have coefficients that are perfect cubes, or even ratios of perfect cubes eg x3 + (8/27)y3.
125,216,343,512
7 cubed is 343