Number of composite numbers around n^n.
0, 1, 5, 5, 15, 13, 5, 45, 19, 51, 103, 53, 27, 43, 79, 71, 91, 171, 19, 141, 33, 109, 133, 129, 97, 105, 77, 173, 305, 25, 131, 53, 257, 115, 77, 49, 89, 447, 65, 213, 301, 139, 351, 465, 245, 669, 593, 395, 19, 243, 227, 639, 545, 461, 353, 1039, 407, 175
1
Although n^n > n! for n > 1, the number of composite numbers around n^n can be much lower.
T. D. Noe, Plot of 500 terms
T. D. Noe, Table of 500 terms
(Mma) Join[{0,}, Table[NextPrime[n^n] - NextPrime[n^n, -1] - 1, {n, 2, 50}]]
nonn
T. D. Noe, Mar 20 2015