A002068 - OEIS

A002068

Wilson remainders: a(n) = ((p-1)!+1)/p mod p, where p = prime(n).
(Formerly M3728 N1524)

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

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.

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

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 *)

PROG

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