A curious case of $1729$

Question

Ramanujan's Taxicab number 1729 is famous for being the smallest positive integer which can be written as the sum of two positive cubes in two different ways. On a different note, I observed that

$12^3 + 1^3 = 1729$

$12^2 + 1^2 = 1\cdot7\cdot2\cdot9 + 1 + 7 + 2 + 9$

$12^1 + 1^1 = -1+7-2+9$

Question 1: Is there any other $n$ number with the property that if $n = a^3 + b^3$ for some positive $a$ and $b$ then

$$ a^2 + b^2 = \text{Products of the digits of $n$} \ + \ \text{Sum of the digits of $n$} $$

Question 2: While searching for a solution of Question 1, the program run by Peter has found only two solution, $(6,11)$ and $(1,12)$ for $1 \le a,b \le 20000$. Looks like there are no more solution. Can this be proven or disproven?

Answer 1

If I understood correctly the problem is

Find numbers $n$ such that there exist a pair $(a,b)$ with the property that

$a^3+b^3=n$

$a^2+b^2$=products of the digits of n + Sum of the digits of n

Then, the number $1547$ is a solution, for the pair $(a,b)=(11,6)$:

$11^3+6^3=1547\\11^2+6^2=1\cdot 5 \cdot 4 \cdot 7 + 1 + 5 + 4 + 7\\11^1+6^1 = 1+5+4+7$

And of course also the number $0$ is a solution. I also feel that there are no more solutions.

Answer 2

The first two conditions are satisfied by $$a=6\ \ \ \ , \ \ \ \ b=11$$

Answer 3

$(a,b)=(12,1)$ and $(a,b)=(11,6)$ are the only integer solutions with $2500 \geq a > b$.