Another variant - read about function PowersRepresentations[n,3,2],
for example
PowersRepresentations[110011, 3, 2]
{{3, 21, 331}, {7, 159, 291}, {7, 219, 249}, {9, 83, 321}, {9, 207, 259}, {15, 15, 331}, {15, 169, 285}, {21, 201, 263}, {27, 79,
321}, {27, 141, 299}, {29, 117, 309}, {29, 177, 279}, {45, 219, 245}, {47, 69, 321}, {47, 111, 309}, {57, 61, 321}, {61, 147,
291}, {63, 129, 299}, {63, 219, 241}, {65, 81, 315}, {69, 69, 317}, {69, 135, 295}, {69, 155, 285}, {79, 171, 273}, {81, 137,
291}, {81, 213, 241}, {83, 159, 279}, {101, 123, 291}, {101, 159, 273}, {111, 219, 223}, {117, 171, 259}, {129, 129, 277}, {137, 171,
249}, {141, 201, 223}, {155, 195, 219}, {159, 169, 237}, {169, 183, 219}, {171, 171, 227}}