Is there a name for this type of logical fallacy? Consider a statement of the form: $A$ implies $B$, where $A$ and $B$ are true, but $B$ is not implied by $A$. 
Example: As $3$ is odd, $3$ is prime. 
In this case, it is true that $3$ is odd, and that $3$ is prime, but the implication is false. If $9$ had been used instead of $3$, the first statement would be true, but the second wouldn't, in which case it is clear that the implication is false.

Is there a name for this sort of logical fallacy?


This is not strictly a fallacy; this is a result from the truth-table definition of implication, in which a statement with a true conclusion is necessarily given the truth-value true, i.e., for any formula $A\rightarrow B$ , if B is given the truth-value T, then, by the truth-table definition of $ \rightarrow $, it follows that $A\rightarrow B$  is true. The only invalid statement of this sort is a statement of the sort
$A\rightarrow B$ in which A is true and B is false; this is the essence of what logic is about; we want to avoid starting with a true statement, arguing correctly, and ending up with a false statement; logic is largely designed to avoid this, as correct reasoning is that which preserves true premises, or sends true premises to true premises.
