The sequence 
is given by 1, 1, 2, 3, 5, 7, 11, 17, 25, 38, ... (OEIS A002379).
The first few composite 
occur for 
,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, ... (OEIS A046037),
corresponding to the composites 25, 38, 57, 86, 129, 194, 291, 437, 656, ... (OEIS
A070758). Similarly, the first few prime 
occur for 
, 4, 5, 6, 7, 21, 22, 98, ... (OEIS A070759),
corresponding to the primes 2, 3, 5, 7, 11, 17, 4987, 7481, 180693856682317883, ...
(OEIS A067904).
The sequence 
is given by 1, 1, 2, 3, 4, 5, 7, 9, 13, 17, 23, ... (OEIS A064628).
The first few composite 
occur for 
,
8, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, ... (OEIS A046038),
corresponding to composites 4, 9, 42, 56, 74, 99, 133, 177, 236, ... (OEIS A070761).
Similarly, the first few prime 
occur for 
, 6, 7, 9, 10, 11, 12, 38, 42, 59, 96,... (OEIS A070762),
corresponding to the primes 2, 3, 5, 7, 13, 17, 23, 31, 55933, 176777, 23517191,
... (OEIS A067905).
There are infinitely many integers of the form 
and 
which are composite, where 
is the floor function
(Forman and Shapiro, 1967; Guy 1994, p. 220).
