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).