What is an integer if half of it is a complete square number and one third of it is a complete cubic number

What is an integer if half of it is a complete square number and one third of it is a complete cubic number

Let this integer be 6 α. ① 6 α △ 2 = 3 α is a complete square number, α can be expressed as α = 3A & # 178; ② 6 α △ 3 = 2 α is a complete cubic number, α can be expressed as α = 2 & # 178; × B & # 179;. ③ α = 3A & # 178; = 2 & # 178; × B & # 179; B is a multiple of 3, B is at least 3. α is at least α = 2 & # 178

Write an integer that is greater than 100 and can be divided by 3, find the cubic sum of the numbers on each digit of the number, repeat the operation, and find out what rules, give an example
Take an example

These numbers are also divisible by three

Write an integer greater than 100 that can be divided by 3, find the cubic sum of the numbers on each digit of the number, and the result will be
Arbitrarily write a number greater than 100 that can be divided by 3, find the cubic sum of the numbers on each digit of the number, and repeat the above operation for the sum. What's the rule?
For example, I can't work out 153 after a long time

For example, for 3213 ^ 3 + 2 ^ 3 + 1 ^ 3 = 27 + 8 + 1 = 36, for 363 ^ 3 + 6 ^ 3 = 27 + 216 = 243, for 2432 ^ 3 + 4 ^ 3 + 3 ^ 3 = 8 + 64 + 27 = 99, for 999 ^ 3 + 9 ^ 3 = 1458, for 14581 ^ 3 + 4 ^ 3 + 5 ^ 3 + 8 ^ 3 = 1 + 64 + 125 + 512 =