Let's call an infinite sequence of bits $f:N\rightarrow \{0,1\}$ *absolutely random* if any computably constructed subsequence is not computable, i.e. there aren't monotonic computable function $g:N \rightarrow N$ and computable function $h:N \rightarrow \{0,1\}$ such that $\forall n ~f(g(n)) = h(n)$.


Is it known definition? I'm sure it's not the same as Martin-Löf randomness, because Chaitin's constant is not absolutely random (we can construct infinite computable sequence of programs that are halting). 
EDIT: Looks like I was wrong and Chaitin's constant in fact is absolutely random.


How to prove that such function exists (or it's not)?



Absolutely algorithmically random infinite sequence
As Liang Yu pointed out, this notion is known as bi-immunity. Chaitin's constant is in fact bi-immune, as is any Martin-Löf random real. Indeed, bi-immunity is much weaker than Martin-Löf randomness. Again as pointed out by Liang Yu, quite weak notions of randomness and (computability theoretic) genericity suffice to ensure bi-immunity, so the set of bi-immune reals is both comeager and conull.


