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

2,1

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

Table of n, a(n) for n=2..23.

K. S. Brown, Highly Wilsonian Primes

Igor Naverniouk, C++ program

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]

Prog

(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

Crossrefs

Sequence in context: A168034 A034970 A048417 * A048403 A000519 A088732

Adjacent sequences: A071707 A071708 A071709 * A071711 A071712 A071713

Keyword

hard,more,nonn

Author

Benoit Cloitre, Jun 03 2002

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

Status

approved