A238501 Primes p for which x! + (p-1)!/x!==0 (mod p) has only three solutions 1<=x<=p-2.

7, 11, 19, 31, 43, 47, 107, 127, 131, 151, 163, 167, 179, 191, 211, 223, 263, 283, 347, 367, 443, 487, 491, 523, 547, 587, 643, 659, 751, 827, 839, 911, 1039, 1051, 1087, 1103, 1123, 1163, 1171, 1223, 1259, 1283, 1291, 1327, 1427, 1439, 1447, 1487, 1523, 1543

Offset: 1

Author

Vladimir Shevelev, Feb 27 2014

Keywords

• nonn

Comments

All terms are of the form 4*k+3.

Using Wilson's theorem, for every p>3, p==3(mod 4) we have, at least, 3 solutions in [1,p-2] of x! + (p-1)!/x!==0 (mod p): x = 1, x = (p-1)/2, x = p-2.

Crossrefs

Cf. A238444, A238460.

Programs

Mathematica

kmax = 400; Select[Select[4*Range[kmax]+3, PrimeQ], (r = Range[#-2]; Count[r!+(#-1)!/r!, k_ /; Divisible[k, #]] == 3)&] (* Jean-François Alcover, Mar 05 2014 *)

Formula

a(n) is prime(k(n)) for which A238444(k(n)) = 3.

Extensions

More terms from Peter J. C. Moses, Feb 27 2014