A007619 -id:A007619 - cp's OEIS frontend

A163212 Wilson quotients (A007619) which are primes.

5, 103, 329891, 10513391193507374500051862069

Offset: 1

Peter Luschny, Jul 24 2009

a(5) = A007619(137), a(6) = A007619(216), a(7) = A007619(381).

Same as A122696 without its initial term 2. - Jonathan Sondow, May 19 2013

			The quotient (720+1)/7 = 103 is a Wilson quotient and a prime, so 103 is a member.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113 [math.NT], 2011-2012.
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.

Cf. A050299, A163211, A007619, A122696, A163210, A163213, A163209, A225906.

# WQ defined in A163210.
A163212 := n -> select(isprime,WQ(factorial,p->1,n)):

Select[Table[p = Prime[n]; ((p-1)!+1)/p, {n, 1, 15}], PrimeQ] (* Jean-François Alcover, Jun 28 2013 *)

forprime(p=2, 1e4, a=((p-1)!+1)/p; if(ispseudoprime(a), print1(a, ", "))) \\ Felix Fröhlich, Aug 03 2014

a(n) = A122696(n+1) = A007619(A225906(n)) = ((A050299(n+1)-1)!+1)/A050299(n+1). - Jonathan Sondow, May 19 2013

A319025 Primes p such that W_p == 2 (mod p), where W_p = A007619(n) and p = prime(n).

Original entry on oeis.org

19, 1187, 14296621, 16556218163369

Offset: 1

Felix Fröhlich, Sep 08 2018

These are the members of René Gy's set W_2 (cf. Gy, 2018).

The sequence is complete to 2*10^13, with the higher terms coming from a list of primes with small Wilson quotients in the article by Costa, Gerbicz, and Harvey. - John Blythe Dobson, Jan 05 2021

Edgar Costa, Robert Gerbicz, and David Harvey, A search for Wilson primes, Mathematics of Computation 83 (2014) 3071-3091.
R. Gy, Generalized Lerch Primes, Integers: Electronic Journal of Combinatorial Number Theory 18, (2018), #A10.

Cf. A007540, A197632.

forprime(p=1, , if(Mod(((p-1)!+1)/p, p)==2, print1(p, ", ")))

A007540 Wilson primes: primes p such that (p-1)! == -1 (mod p^2).

Original entry on oeis.org

5, 13, 563

Offset: 1

N. J. A. Sloane

Suggested by the Wilson-Lagrange Theorem: An integer p > 1 is a prime if and only if (p-1)! == -1 (mod p). Cf. Wilson quotients, A007619.
Sequence is believed to be infinite. Next term is known to be > 2*10^13 (cf. Costa et al., 2013).
Intersection of the Wilson numbers A157250 and the primes A000040. - Jonathan Sondow, Mar 04 2016
Conjecture: Odd primes p such that 1^(p-1) + 2^(p-1) + ... + (p-1)^(p-1) == p-1 (mod p^2). - Thomas Ordowski and Giovanni Resta, Jul 25 2018
From Felix Fröhlich, Nov 16 2018: (Start)
Harry S. Vandiver apparently said about the Wilson primes "It is not known if there are infinitely many Wilson primes. This question seems to be of such a character that if I should come to life any time after my death and some mathematician were to tell me that it had definitely been settled, I think I would immediately drop dead again." (cf. Ribenboim, 2000, p. 217).
Let p be a Wilson prime and let i be the index of p in A000040. For n = 1, 2, 3, the values of i are 3, 6, 103. The primes among those values are Lerch primes, i.e., terms of A197632. Is this a property that necessarily follows if i is prime (cf. Sondow, 2011/2012, 2.5 Open Problems 5)? (End)
From Amiram Eldar, Jun 16 2021: (Start)
Named after the English mathematician John Wilson (1741-1793) after whom "Wilson's theorem" was also named.
The primes 5 and 13 appear in an exercise involving the Wilson congruence in Mathews (1892). [Edited by Felix Fröhlich, Jul 23 2021]
Beeger found that there are no other smaller terms up to 114 (1913) and up to 200 (1930).
a(3) = 563 was found by Goldberg (1953), who used the Bureau of Standards Eastern Automatic Computer (SEAC) to search all primes less than 10000. According to Goldberg, the third prime was discovered independently by Donald Wall six month later. (End)

N. G. W. H. Beeger, On the Congruence (p-1)! == -1 (mod p^2), Messenger of Mathematics, Vol. 49 (1920), pp. 177-178.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 52.
Calvin C. Clawson, Mathematical Mysteries, Plenum Press, 1996, p. 180.
Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 29.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 80.
G. B. Mathews, Theory of Numbers Part I., Cambridge: Deighton, Bell and Co., London: George Bell and Sons, 1892, page 318.
Paulo Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory, Springer Science & Business Media, 2000, ISBN 0-387-98911-0.
Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 277.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 234-235.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Ilan Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 73.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, p. 163.

N. G. W. H. Beeger, Quelques remarques sur les congruences r^(p-1) == 1 (mod p^2) et (p- 1)! == -1 (mod p^2), The Messenger of Mathematics, Vol. 43 (1913), pp. 72-84.
Edgar Costa, Robert Gerbicz and David Harvey, A search for Wilson primes, Mathematics of Computation, Vol. 83, No. 290 (2014), pp. 3071-3091; arXiv preprint, arXiv:1209.3436 [math.NT], 2012.
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation, 66 (1997), 433-449.
Karl Goldberg, A Table of Wilson Quotients and the Third Wilson Prime, Journal of the London Mathematical Society, Vol. 28 (1953), pp. 252-256.
James Grime and Brady Haran, What do 5, 13 and 563 have in common?, YouTube video (2014).
Emma Lehmer, A Note on Wilson's Quotient, The American Mathematical Monthly, Vol. 44, No. 4 (1937), pp. 237-238.
Emma Lehmer, On the Congruence (p-1)! == -1 (mod p^2), The American Mathematical Monthly, Vol. 44, No. 7 (1937), p. 462.
Emma Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson", Annals of Mathematics, Vol. 39, No. 2 (1938), pp. 350-360.
George Ballard Mathews, Theory of numbers, Part I, Cambridge, 1892, p. 318.
Tapio Rajala, Status of a search for Wilson primes
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-Non-Wilson Primes 2, 3, 14771, In: M. B. Nathanson, Combinatorial and Additive Number Theory, Springer, CANT 2011 and 2012. Also on arXiv, arXiv:1110.3113 [math.NT], 2011-2012.
Apoloniusz Tyszka, On sets X subset of N for which we know an algorithm that computes a threshold number t(X) in N such that X is infinite if and only if X contains an element greater than t(X), 2019.
Eric Weisstein's World of Mathematics, Wilson Prime.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.
Wikipedia, Wilson prime.
Paul Zimmermann, Records for prime numbers.

Cf. A007619, A157249, A157250, A377266.

Select[Prime[Range[500]], Mod[(# - 1)!, #^2] == #^2 - 1 &] (* Harvey P. Dale, Mar 30 2012 *)

forprime(n=2, 10^9, if(Mod((n-1)!, n^2)==-1, print1(n, ", "))) \\ Felix Fröhlich, Apr 28 2014

is(n)=prod(k=2,n-1,k,Mod(1,n^2))==-1 \\ Charles R Greathouse IV, Aug 03 2014

from sympy import prime
A007540_list = []
for n in range(1,10**4):
    p, m = prime(n), 1
    p2 = p*p
    for i in range(2,p):
        m = (m*i) % p2
    if m == p2-1:
        A007540_list.append(p) # Chai Wah Wu, Dec 04 2014

A089026 a(n) = n if n is a prime, otherwise a(n) = 1.

Original entry on oeis.org

1, 2, 3, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 1, 1, 1, 17, 1, 19, 1, 1, 1, 23, 1, 1, 1, 1, 1, 29, 1, 31, 1, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 1, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 1, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 1, 1, 83, 1, 1, 1, 1, 1, 89, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Roger L. Bagula, Nov 12 2003

nonn

This sequence was the subject of the 1st problem of the 9th Irish Mathematical Olympiad 1996 with gcd((n + 1)!, n! + 1) = a(n+1) for n >= 0 (see formula Jan 23 2009 and link). - Bernard Schott, Jul 22 2020

For sequence A with terms a(1), a(2), a(3),... , let R(0) = 1 and for k >= 1 let R(k) = rad(a(1)*a(2)*...*a(k)). Define the Rad-transform of A to be R(n)/R(n-1); n >= 1, where rad is A007947. Then this sequence is the Rad transform of the positive integers, A = A000027. - David James Sycamore, Apr 19 2024

			From Larry Tesler (tesler(AT)pobox.com), Nov 08 2010: (Start)
a(9) = (8*9*10)/(2^((5+2+1)-(3+1+0))*3^((3+1)-(2+0))*5^((2)-(1))*7^((1)-(1))) = 1 [composite].
a(10) = (8*9*10)/(2^((5+2+1)-(3+1+0))*3^((3+1)-(2+0))*5^((2)-(1))*7^((1)-(1))) = 1 [composite].
a(11) = (8*9*10*11*12)/(2^((6+3+1)-(3+1+0))*3^((4+1)-(2+0))*5^((2)-(1))*7^((1)-(1))) = 11 [prime]. (End)

Paulo Ribenboim, The little book of big primes, Springer 1991, p. 106.
L. Tesler, "Factorials and Primes", Math. Bulletin of the Bronx H.S. of Science (1961), 5-10. [From Larry Tesler (tesler(AT)pobox.com), Nov 08 2010]

G. C. Greubel, Table of n, a(n) for n = 1..5000
The IMO Compendium, Problem 1, 9th Irish Mathematical Olympiad 1996.
Index to sequences related to Olympiads.

Differs from A080305 at n=30.
Cf. A090585, A000217, A069268, A090586, A007619, A007406.
Cf. A061397, A135683.
Cf. A000027, A007947.

a = [1:96]; a(isprime(a) == false) = 1; % Thomas Scheuerle, Oct 06 2022

[IsPrime(n) select n else 1: n in [1..96]]; // Marius A. Burtea, Aug 02 2019

digits=200; a=Table[If[PrimePi[n]-PrimePi[n-1]>0, n, 1], {n, 1, digits}]; Table[Numerator[(n/2)/(n-1)! ] + Floor[2/n] - 2*Floor[1/n], {n,1,200}] (* Alexander Adamchuk, May 20 2006 *)
Range@ 120 /. k_ /; CompositeQ@ k -> 1 (* or *)
Table[n Boole@ PrimeQ@ n, {n, 120}] /. 0 -> 1 (* or *)
Table[If[PrimeQ@ n, n, 1], {n, 120}] (* Michael De Vlieger, Jul 02 2016 *)

a(n) = n^isprime(n) \\ David A. Corneth, Oct 06 2022

from sympy import isprime
def a(n): return n if isprime(n) else 1
print([a(n) for n in range(1, 97)]) # Michael S. Branicky, Oct 06 2022

def A089026(n):
    if n == 4: return 1
    f = factorial(n-1)
    return (f + 1) - n*(f//n)
[A089026(n) for n in (1..96)]   # Peter Luschny, Oct 16 2013

From Peter Luschny, Nov 29 2003: (Start)
a(n) = denominator(n! * Sum_{m=0..n} (-1)^m*m!*Stirling2(n+1, m+1)/(m+1)).
a(n) = denominator(n! * Sum_{m=0..n} (-1)^m*m!*Stirling2(n, m)/(m+1)). (End)
From Alexander Adamchuk, May 20 2006: (Start)
a(n) = numerator((n/2)/(n-1)!) + floor(2/n) - 2*floor(1/n).
a(n) = A090585(n-1) = A000217(n-1)/A069268(n-1) for n>2. (End)
a(n) = gcd(n,(n-1)!+1). - Jaume Oliver Lafont, Jul 17 2008, Jan 23 2009
a(1) = 1, a(2) = 2, then a(n) = 1 or a(n) = n = prime(m) = (Product q+k, k = 1 .. 2*floor(n/2+1)-q) / (Product prime(i)^(Sum (floor((n+1)/(prime(i)^w)) - floor(q/(prime(i)^w)) ), w = 1 .. floor(log[base prime(i)] n+1) ), i = 2 .. m-1) where q = prime(m-1). -  Larry Tesler (tesler(AT)pobox.com), Nov 08 2010
a(n) = (n!*HarmonicNumber(n) mod n)+1, n != 4. - Gary Detlefs, Dec 03 2011
a(n) = denominator of (n!)/n^(3/2). - Arkadiusz Wesolowski, Dec 04 2011
a(n) = A034386(n+1)/A034386(n). - Eric Desbiaux, May 10 2013
a(n) = n^c(n), where c = A010051. - Wesley Ivan Hurt, Jun 16 2013
a(n) = A014963(n)^(-A008683(n)). - Mats Granvik, Jul 02 2016
Conjecture: for n > 3, a(n) = gcd(n, A007406(n-1)). - Thomas Ordowski, Aug 02 2019
a(n) = 1 + c(n)*(n-1), where c = A010051. - Wesley Ivan Hurt, Jun 21 2025

A000940 Number of n-gons with n vertices.

Original entry on oeis.org

1, 2, 4, 12, 39, 202, 1219, 9468, 83435, 836017, 9223092, 111255228, 1453132944, 20433309147, 307690667072, 4940118795869, 84241805734539, 1520564059349452, 28963120073957838, 580578894859915650, 12217399235411398127, 269291841184184374868, 6204484017822892034404

Offset: 3

N. J. A. Sloane

Number of inequivalent undirected Hamiltonian cycles in complete graph on n labeled nodes under action of dihedral group of order 2n acting on nodes.

			Label the vertices of a regular n-gon 1,2,...,n.
For n=3,4,5 representatives for the polygons counted here are:
  (1,2,3,1),
  (1,2,3,4,1), (1,2,4,3,1),
  (1,2,3,4,5,1), (1,2,3,5,4,1), (1,2,4,5,3,1), (1,3,5,2,4,1).
For n=6:
  (1,2,3,4,5,6,1), (1,2,3,4,6,5,1), (1,2,3,5,6,4,1),
  (1,2,3,6,5,4,1), (1,2,4,3,6,5,1), (1,2,4,6,3,5,1),
  (1,2,4,6,5,3,1), (1,2,5,3,6,4,1), (1,2,5,4,6,3,1),
  (1,2,5,6,3,4,1), (1,2,6,4,5,3,1), (1,3,5,2,6,4,1).

J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161.
R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
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).

Ludovic Schwob, Table of n, a(n) for n = 3..500 (terms 3..100 from T. D. Noe).
S. W. Golomb and L. R. Welch, On the enumeration of polygons, Amer. Math. Monthly, 67 (1960), 349-353.
S. W. Golomb and L. R. Welch, On the enumeration of polygons, Amer. Math. Monthly, 67 (1960), 349-353. [Annotated scanned copy]
Samuel Herman and Eirini Poimenidou, Orbits of Hamiltonian Paths and Cycles in Complete Graphs, arXiv:1905.04785 [math.CO], 2019.
E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143.
R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971 (gives initial terms of this sequence, except he has a(6)=7 instead of 12)
R. C. Read, Letter to N. J. A. Sloane, 1992
R. C. Read, Combinatorial problems in theory of music, Discrete Math. 167 (1997), 543-551.
Ludovic Schwob, On the enumeration of double cosets and self-inverse double cosets, arXiv:2506.04007 [math.CO], 2025. See p. 9.
N. J. A. Sloane, Illustration of initial terms [Annotated page from Golomb-Welch article]
Venta Terauds and J. Sumner, Circular genome rearrangement models: applying representation theory to evolutionary distance calculations, arXiv preprint arXiv:1712.00858 [q-bio.PE], 2017.
Venta Terauds and Jeremy Sumner, A new algebraic approach to genome rearrangement models, arXiv:2012.11665 [q-bio.PE], 2020.

Cf. A000939, A007619. Bisections give A094156, A094157.

For permutation classes under various symmetries see A089066, A262480, A002619.

with(numtheory);
# for n odd:
Sd:=proc(n) local t1,d; t1:=2^((n-1)/2)*n^2*((n-1)/2)!; for d from 1 to n do if n mod d = 0 then t1:=t1+phi(n/d)^2*d!*(n/d)^d; fi; od: t1/(4*n^2); end;
# for n even:
Se:=proc(n) local t1,d; t1:=2^(n/2)*n*(n+6)*(n/2)!/4; for d from 1 to n do if n mod d = 0 then t1:=t1+phi(n/d)^2*d!*(n/d)^d; fi; od: t1/(4*n^2); end;
A000940:=n-> if n mod 2 = 0 then Se(n) else Sd(n); fi;

a[n_] := (t1 = If[OddQ[n], 2^((n - 1)/2)*n^2*((n - 1)/2)!, 2^(n/2)*n*(n + 6)*(n/2)!/4]; For[ d = 1 , d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[n/d]^2*d!*(n/d)^d]]; t1/(4*n^2)); Table[a[n], {n, 3, 25}] (* Jean-François Alcover, Jun 19 2012, after Maple *)

a(n)={if(n<3, 0, (2^(n\2-2)*(n\2)!*n*if(n%2, 4*n, n + 6) + sumdiv(n, d, eulerphi(n/d)^2*d!*(n/d)^d))/(4*n^2))} \\ Andrew Howroyd, Sep 09 2018

from sympy import factorial, divisors, totient
def A000940(n): return 1 if n == 3 else ((sum(totient(m:=n//d)**2*factorial(d)*m**d for d in divisors(n,generator=True))+(1<<(k:=n>>1)-2)*n*(n<<2 if n&1 else (n+6))*factorial(k))>>2)//n//n # Chai Wah Wu, Nov 07 2022

For formula see Maple lines.
a(p) = ((((p-1)! + 1)/p) + p - 2 + (2^((p-1)/2)*((p-1)/2)!))/4 for prime p. See A007619. - Ian Mooney, Oct 05 2022
a(n) ~ sqrt(2*Pi)/4 * n^(n-3/2) / e^n. - Ludovic Schwob, Nov 03 2022

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 05 2004

A238693 Quotients connected with the Banach matchboxes problem: Sum_{i=1..prime(n)-5} 2^(i-1)*binomial(i+1,2)/prime(n) (case 2).

Original entry on oeis.org

0, 1, 93, 571, 16143, 79333, 1755225, 160251339, 705725473, 57691858003, 1057609507815, 4500326662525, 80662044522801, 5995948088798691, 437230824840308493, 1820340203482736875, 130228506669621162901, 2230237339841166071433, 9214275012380069727751

Offset: 3

Vladimir Shevelev, Mar 03 2014

nonn

A general congruence connected with the Banach matchboxes problem is the following: for k=1,2,...,(p-1)/2, Sum_{i=1..p-2k-1} 2^(i-1)*binomial(k-1+i,k) == 0 (mod p) (p is odd prime). If k=1 (case 1), then one can prove that the corresponding quotients are 2^(prime(n)-3) - A007663(n), n >= 2.

Vladimir Shevelev, Banach matchboxes problem and a congruence for primes, arXiv:1110.5686 [math.HO], 2011.

Cf. A007663, A007619, A238692.

Array[Sum[2^(i - 1)*Binomial[i + 1, 2]/#, {i, # - 5}] &@ Prime@ # &, 19, 3] (* Michael De Vlieger, Dec 06 2018 *)

a(n) = sum(i=1, prime(n)-5, 2^(i-1)*binomial(i+1,2))/prime(n); \\ Michel Marcus, Dec 06 2018

More terms from Peter J. C. Moses, Mar 03 2014

A050299 Numbers k such that ((k-1)! + 1)/k is prime.

Original entry on oeis.org

1, 5, 7, 11, 29, 773, 1321, 2621

Offset: 1

N. J. A. Sloane, Apr 09 2003

Except for the first term, all terms are primes because for n > 1, n divides (n-1)! + 1 iff n is prime. - Farideh Firoozbakht, Mar 19 2004

a(9) >= 30941.

			7 is in the sequence because (6!+1)/7=103 is prime.

Mike Oakes, posting to Number Theory List, Aug 20 2003
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113 [math.NT], 2011-2012.
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.
Javier Soria, posting to Number Theory List, Apr 08 2003

v={1};Do[If[PrimeQ[((Prime[n]-1)!+1)/Prime[n]], v=Append[v, Prime[n]];Print[v]], {n, 845}]
Select[Range[2630],PrimeQ[((#-1)!+1)/#]&] (* Harvey P. Dale, Aug 18 2024 *)

is(n)=((n-1)!+1)%n==0 && isprime(((n-1)!+1)/n) \\ Anders Hellström, Nov 22 2015

((a(n)-1)! + 1)/a(n) = A122696(n) = A007619(A000720(A050299(n))) for n > 1. - Jonathan Sondow, Aug 07 2011

a(n) = prime(A225906(n-1)) for n > 1. - Jonathan Sondow, May 20 2013

a(7)-a(8) from Mike Oakes, Aug 20 2003

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

Original entry on oeis.org

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

N. J. A. Sloane

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

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).

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.

Cf. A007540, A007619, A275741.

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

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 *)

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

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

A163210 Swinging Wilson quotients ((p-1)$ +(-1)^floor((p+2)/2))/p, p prime. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

1, 1, 1, 3, 23, 71, 757, 2559, 30671, 1383331, 5003791, 245273927, 3362110459, 12517624987, 175179377183, 9356953451851, 509614686432899, 1938763632210843, 107752663194272623

Offset: 1

Peter Luschny, Jul 24 2009

nonn

			The 5th prime is 11, (11-1)$ = 252, the remainder term is (-1)^floor((11+2)/2)=1. So the quotient (252+1)/11 = 23 is the 5th member of the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..470
M. E. Bassett, S. Majid, Finite noncommutative geometries related to F_p[x], arXiv:1603.00426 [math.QA], 2016.
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A163213, A002068, A163212, A163209, A007619, A007540.

swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
WQ := proc(f,r,n) map(p->(f(p-1)+r(p))/p,select(isprime,[$1..n])) end:
A163210 := n -> WQ(swing,p->(-1)^iquo(p+2,2),n);

sf[n_] := n!/Quotient[n, 2]!^2; a[n_] := (p = Prime[n]; (sf[p - 1] + (-1)^Floor[(p + 2)/2])/p); Table[a[n], {n, 1, 19}] (* Jean-François Alcover, Jun 28 2013 *)
a[p_] := (Binomial[p-1, (p-1)/2] - (-1)^((p-1)/2)) / p
Join[{1, 1}, a[Prime[Range[3,20]]]] (* Peter Luschny, May 13 2017 *)

a(n, p=prime(n)) = ((p-1)!/((p-1)\2)!^2 - (-1)^(p\2))/p \\ David A. Corneth, May 13 2017

A197636 Non-Wilson primes: primes p such that (p-1)! =/= -1 mod p^2.

Original entry on oeis.org

2, 3, 7, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 569

Offset: 1

Jonathan Sondow, Oct 19 2011

nonn

All primes except 5, 13, 563, and any other Wilson prime A007540 that may exist.
Same as primes p that do not divide their Wilson quotient ((p-1)!+1)/p.
Wilson's theorem says that (p-1)! == -1 (mod p) if and only if p is prime.
p = prime(i) is a term iff A250406(i) != 0. - Felix Fröhlich, Jan 24 2016
Complement of A007540 in A000040. - Felix Fröhlich, Jan 24 2016

			2 is a non-Wilson prime since (2-1)! = 1 ==/== -1 (mod 2^2).

E. Costa, R. Gerbicz and D. Harvey, A search for Wilson primes, Mathematics of Computation, 83 (2014), 3071-3091 (arXiv:1209.3436 [math.NT], 2012).
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113 [math.NT], 2011-2012.
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.

Cf. A007540, A007619, A197633, A197634, A197635, A250406.

Select[Prime@ Range@ 104, Mod[Factorial[# - 1], #^2] != #^2 - 1 &] (* Michael De Vlieger, Jan 24 2016 *)

forprime(p=1, 500, if(Mod((p-1)!, p^2)!=-1, print1(p, ", "))) \\ Felix Fröhlich, Jan 24 2016

((p-1)!+1)/p =/= 0 (mod p), where p is prime.

cp's OEIS Frontend

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A007540 Wilson primes: primes p such that (p-1)! == -1 (mod p^2).

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A089026 a(n) = n if n is a prime, otherwise a(n) = 1.

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A000940 Number of n-gons with n vertices.

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A238693 Quotients connected with the Banach matchboxes problem: Sum_{i=1..prime(n)-5} 2^(i-1)*binomial(i+1,2)/prime(n) (case 2).

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