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A002068 Wilson remainders: a(n) = ((p-1)!+1)/p mod p, where p = prime(n).
(Formerly M3728 N1524)
8
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
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.
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
CROSSREFS
Sequence in context: A021200 A265301 A019904 * A021666 A340841 A143148
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved

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Last modified March 1 11:49 EST 2024. Contains 370432 sequences. (Running on oeis4.)