A071710 Highly Wilsonian primes: smallest primes p such that w(p)=n where w(n) denote the number of nonnegative integers k such that k! = +1 or -1 (mod n).

2, 3, 5, 7, 17, 67, 137, 23, 61, 71, 401, 1907, 661, 12227, 29873, 96731, 99721, 154243, 480209, 3408707, 1738901, 27341387

Offset: 2

Author

Benoit Cloitre, Jun 03 2002

Keywords

• hard
• more
• nonn

Comments

Obviously w(n) is at least 2 because 0! = 1! = +1 (mod n) for every n. Also, if p is a prime, then w(p) is at least 4 because (p-2)! = +1 and (p-1)! = -1 (mod p) by Wilson's Theorem.

The smallest prime(k) such that A238444(k) = n-2. - Vladimir Shevelev, Feb 28 2014

The sequence w(n) is 1, 2, 3, 2, 4, 2, 5, 2, 2, 2, 5, 2, 4,... (offset 1) = 1 +A049046(n) +A238532(n) for n>2. - R. J. Mathar, Apr 02 2014

Links

• K. S. Brown, Highly Wilsonian Primes
• Igor Naverniouk, C++ program

Programs

Mathematica

w[n_] := Block[{c = k = m = 1}, While[k < n, m = Mod[m *= k, n]; If[m == 1 || m + 1 == n, c++ ]; k++ ]; c]

PARI

wilsonian(p)={ local(s,t,pMinusOne); pMinusOne=p-1; s=4; t=24; for(k=5,p-3, t=(t*k)%p; if(t==1 || t==pMinusOne, s=s+1) ); s } \\ Charles R Greathouse IV, Jan 24 2007

Extensions

2 more terms from Charles R Greathouse IV, Jan 24 2007

a(23) from Igor Naverniouk (igor(AT)cs.utoronto.ca), May 09 2007