A049046 -id:A049046 - cp's OEIS frontend

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

Benoit Cloitre, Jun 03 2002

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

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

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

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

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

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

A049044 Least k > 0 such that A049045(n) | A003422(k-1).

Original entry on oeis.org

1, 2, 3, 4, 6, 4, 6, 6, 5, 7, 6, 7, 12, 5, 22, 7, 16, 7, 55, 12, 54, 42, 22, 6, 16, 24, 25, 86, 55, 97, 133, 54, 42, 6, 64, 94, 72, 58, 24, 49, 69, 19, 25, 78, 86, 14, 208, 167, 138, 80, 97, 59, 133, 63, 142, 41, 110, 64, 22, 94, 286, 72, 39, 58, 84, 215, 80, 14, 49, 305, 69

Offset: 1

David W. Wilson

nonn

Cf. A049041, A049045, A049046, A003422.

A238532 Number of distinct factorial numbers congruent to -1 (mod n).

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 3, 0, 0, 0, 7, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 3, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

R. J. Mathar, Apr 02 2014

nonn

Number of solutions to k! == -1 (mod n), k>=1.
Counts the frequency of the value n-1 in the n-th row of triangle A062169.
Values 1..9 occur for the first time at n = 2, 7, 23, 59, 227, 401, 71, 3643, 62939, which are all prime numbers (see also A230315). Sequence A256519 gives composite k for which a(k) > 0. - Antti Karttunen, May 24 2021

			There are two 6's in the 7th row of A062169. Therefore a(7)=2.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to factorial numbers

Cf. A000142, A049046, A062169, A230315, A256519.

A238532 := proc(n)
    local a,k ;
    a := 0 ;
    for k from 1 to n-1 do
        if modp(k!,n) = modp(-1,n) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Apr 02 2014

A238532(n) = { my(m=1,s=0); for(k=1,oo,m *= k; if(!(m%n),return(s), if(1+(m%n)==n, s++))); }; \\ Antti Karttunen, May 24 2021

A238532(n) = { my(m=Mod(1,n),s=0,x); for(k=1,oo, m *= Mod(k,n); x = lift(m); if(!x,return(s), if(x==(n-1), s++))); }; \\ (Much faster than above program) - Antti Karttunen, May 24 2021

A049050 Least p with exactly n distinct factorial numbers == 1 (mod p).

Original entry on oeis.org

2, 5, 29, 17, 23, 199, 619, 3313, 4093, 3011, 355799, 52163, 12087071, 3408707, 26927543

Offset: 1

David W. Wilson

Cf. A049046, A071710.

a(13)-a(15) from Giovanni Resta, Jun 28 2019

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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).

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A049044 Least k > 0 such that A049045(n) | A003422(k-1).

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A238532 Number of distinct factorial numbers congruent to -1 (mod n).

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A049050 Least p with exactly n distinct factorial numbers == 1 (mod p).

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