A set 
of integers is productive if there exists a partial recursive function 
such that, for any 
, the following holds: If the domain of 
is a subset of 
, then 
is convergent, 
belongs to 
, and 
does not belong to the domain of 
, where 
denotes a recursive function whose Gödel
number is 
.
Productive sets are not recursively enumerable.
