Not computable but left computable number

computability

I can't find an example of number that is not computable but it is left computable.
In general is already difficult to give example of non computable numbers, but I can't even find any number which is not left computable.

Any help is appreciated.

P.S:
A real number $$x$$, is said to be left computable number if the set $$\{q\in\mathbb{Q}\vert q is recursively enumerable (that is, it is the image of a Turing machine).

Consider a sequence $$q_{n}$$ subunitary rational numbers.
Define the $$k$$'th digit after the decimal point of $$q_{n}$$ to be $$1$$ if $$k \leq n$$ and the program encoded by $$k$$ halts after $$n$$ steps or less, or $$0$$ otherwise.
The $$q_{n}$$ are increasing, as a digit that becomes $$1$$ stays $$1$$ for all $$q_{p}$$ with $$p>n$$. They are also rational, as each $$q_{n}$$ has at most $$n$$ non-zero decimal places. However, their limit will be a number $$q$$ who's $$k$$th digit is $$1$$ if the program encoded by $$k$$ halts, or $$0$$ otherwise. As it encodes the solution to the halting problem, it is not computable.
Thus $$q$$ is left-computable, but not computable.