Interestingly of the four 3-digit numbers with property 1A, 370 and 371 are consecutive (407 being the 4th number that is the sum of the cube of its digits).
Argh; I was interpreting (sum of its 3 digits)^3, instead of digit^3 + digit^3 + digit^3. And there is no answer to the former, except 000... and those aren't odd. LOL.