Near-narcissistic number

Since $F$ is a $5$ digit number with all its digits distinct, the maximum value of the digit sum of $F$ can be $(9+8+7+6+5)=35$ .

Since $F$ is the cube of its digit sum, the maximum value that $F$ can achieve is $35^{3}=42875$ . So, our upper bound is $42875$ which is obtained by the maximum possible digit sum.

Now, since $F$ is a $5$ digit number as well as a perfect cube of an integer, the smallest value that $F$ can achieve is $22^{3}=10648$ because this is the smallest $5$ digit perfect cube. So, the lower bound of the digit sum is $22$ and the lowest number is $10648$ .

Now, we are left with finding $F$ such that the digit sum of $F$ lies between $22$ and $35$ inclusive and is equal to the cube of the digit sum. Now, we also know that $F$ lies between $10648$ and $42875$ inclusive.

Checking these $14$ cases, we can easily obtain $F=19683$ when the digit sum is $27$ .

Bonus: The other solution is $17576$ with digit sum $26$ .