A002068 Wilson remainders: a(n) = ((p-1)!+1)/p mod p, where p = prime(n).

1, 1, 0, 5, 1, 0, 5, 2, 8, 18, 19, 7, 16, 13, 6, 34, 27, 56, 12, 69, 11, 73, 20, 70, 70, 72, 57, 1, 30, 95, 71, 119, 56, 67, 94, 86, 151, 108, 21, 106, 48, 72, 159, 35, 147, 118, 173, 180, 113, 131, 169, 107, 196, 214, 177, 73, 121, 170, 25, 277, 164, 231, 271, 259, 288, 110

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

If this is zero, p is a Wilson prime (see A007540).

Costa, Gerbicz, & Harvey give an efficient algorithm for computing terms of this sequence. - Charles R Greathouse IV, Nov 09 2012

References

• R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 29.
• J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 244.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..2000
• Edgar Costa, Robert Gerbicz, and David Harvey, A search for Wilson primes, arXiv:1209.3436 [math.NT], 2012.
• C.-E. Froberg, Investigation of the Wilson remainders in the interval 3<=p<=50,000, Arkiv f. Matematik, 4 (1961), 479-481.
• J. W. L. Glaisher, On the residues of the sums of products of the first p-1 numbers, and their powers, to modulus p^2 or p^3, Quart. J. Math. Oxford 31 (1900), 321-353.
• K. Goldberg, A table of Wilson quotients and the third Wilson prime, J. London Math. Soc., 28 (1953), 252-256.
• J. Sondow, Lerch quotients, Lerch primes, Fermat-Wilson quotients, and the Wieferich-non-Wilson primes 2, 3, 14771, In: Nathanson M. (eds) Combinatorial and Additive Number Theory. Springer Proceedings in Mathematics & Statistics, Vol. 101, Springer, New York, NY, 2014, pp. 243-255, preprint, arXiv:1110.3113 [math.NT], 2011-2012.

Crossrefs

Cf. A007540, A007619, A275741.

Programs

Maple

f:= p -> ((p-1)!+1 mod p^2)/p;
seq(f(ithprime(i)),i=1..1000); # Robert Israel, Jun 15 2014

Mathematica

Table[p=Prime[n]; Mod[((p-1)!+1)/p, p], {n,100}] (* T. D. Noe, Mar 21 2006 *)
Mod[((#-1)!+1)/#,#]&/@Prime[Range[70]] (* Harvey P. Dale, Feb 21 2020 *)

PARI

forprime(n=2, 10^2, m=(((n-1)!+1)/n)%n; print1(m, ", ")) \\ Felix Fröhlich, Jun 14 2014

Formula

a(n) = A007619(n) mod A000040(n).

a(n) + A197631(n) = A275741(n) for n > 1. - Jonathan Sondow, Jul 08 2019

a(n) = ( A027641(p-1)/A027642(p-1) + 1/p - 1 ) mod p, where p = prime(n), proved by Glashier (1900). - Max Alekseyev, Jun 20 2020