|
|
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).
|
|
3
|
|
|
2, 3, 5, 7, 17, 67, 137, 23, 61, 71, 401, 1907, 661, 12227, 29873, 96731, 99721, 154243, 480209, 3408707, 1738901, 27341387
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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 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
|
|
|
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
|
|
|
KEYWORD
|
hard,more,nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
a(23) from Igor Naverniouk (igor(AT)cs.utoronto.ca), May 09 2007
|
|
STATUS
|
approved
|
|
|
|