I read that the following language is r.e. but not not-Turing recognizable
$L$: On input $M$ (where $M$ is a Turing Machine), $M$ accepts at least 20 inputs
I am not sure why it is not not-Turing recognizable., since I could perhaps make the following reduction from $\overline{A_{TM}}$ to $L$ given this procedure $R$ namely:
$R$: On input $<M,w>$:
- Construct TM $M_1$, where on input $x$, if $x=1$, accept
- If input $x$ is not equal to $1$, run $M$ on input $w$ for $|x|$ steps. If after $|x|$ steps, $M$ does not accept $w$, then accept $x$
From this reduction, if $M$ does not accept $w$, i.e. $<M,w> \in \overline{A_{TM}}$, then $M_1$ accepts any input word, i.e. $M_1 \in L$.
Am I missing something here?