Primes p such that p^2 divides k^k + (-1)^k (k-1)^(k-1) for some k > 1.
59, 83, 179, 193, 337, 419, 421, 443, 457, 547, 601, 619, 701, 787, 857, 887, 911, 929, 977, 1039, 1091, 1093, 1109, 1193, 1217, 1223, 1237, 1259, 1289, 1439, 1487, 1489, 1493, 1613, 1637, 1657, 1811, 1847, 1901, 1993, 1997, 2003, 2087, 2089, 2113, 2377, 2389
1
The paper by Boyd et. al. mentions these primes but does not make an effort to find many of them. The sequence A238194 uses these primes to make a list of the k for which k^k + (-1)^k (k-1)^(k-1) is not squarefree.
T. D. Noe, Plot of 153 terms
T. D. Noe, Table of 153 terms
David W. Boyd, Greg Martin, and Mark Thom, Squarefree values of trinomial discriminants, LMS J. Comput. Math. 18 (1) (2015), p. 148-169
(Mma) t1 = {}; Do[s = Select[Range[2, p^2], Mod[PowerMod[#, #, p^2] + (-1)^# PowerMod[# - 1, # - 1, p^2], p^2] == 0 &, 1]; If[Length[s] > 0, AppendTo[t1, p]], {p, Prime[Range[PrimePi[1000]]]}]; t1
nonn,hard
T. D. Noe, Mar 18 2017