A071710 - OEIS

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

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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