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User:Peter Luschny/SwingingPrimes

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Swinging Primes

Factorial primes, primes which are within 1 of a factorial number.
n! A000142
Swinging primes, primes which are within 1 of a swinging factorial number.
n≀ A056040

On this page `?´ is a meta symbol denoting either `!´ or `≀´.
 ! A088054  ≀ A163074

A74 := proc(f,n)
select(isprime,
map(x -> f(x)+1,[$1..n]));
select(isprime,
map(x -> f(x)-1,[$1..n]));
sort(convert(convert(%%,set)
union

convert(%,set),list)) end:
2, 3, 5, 7, 23, 719, 5039, 39916801,
479001599, 87178291199
2, 3, 5, 7, 19, 29, 31, 71, 139, 251, 631, 3433, 12011, 48619, 51479, 51481, 2704157
Primes of the form n? + 1
 ! A088332  ≀ A163075

A75 := proc(f,n)
select(isprime,
map(x -> f(x)+1,[$1..n]))

end:
2, 3, 7, 39916801, 10888869450418352160768000001 2, 3, 7, 31, 71, 631, 3433, 51481, 2704157
Primes of the form n? - 1.
 ! A055490  ≀ A163076

A76 := proc(f,n)
select(isprime,
map(x -> f(x)-1,[$1..n]));

sort(%) end:
5, 23, 719, 5039, 479001599, 87178291199 5, 19, 29, 139, 251, 12011, 48619, 51479, 155117519
Numbers n such that n? + 1 is prime.
 ! A002981  ≀ A163077

A77 := proc(f,n)

select(x -> isprime(f(x)+1),[$0..n]) end:
0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320 0, 1, 2, 3, 4, 5, 8, 9, 14, 15, 24, 27, 31, 38, 44
Numbers n such that n? - 1 is prime.
 ! A002982  ≀ A163078

A78 := proc(f,n)

select(x -> isprime(f(x)-1),[$0..n]) end:
3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324 3, 4, 5, 6, 7, 10, 13, 15, 18, 30, 35, 39, 41, 47
Primes p such that p? + 1 is also prime.
 ! A093804  ≀ A163079

A79 := proc(f,n)
select(isprime,

select(k -> isprime(f(k)+1),[$0..n])) end:
2, 3, 11, 37, 41, 73, 26951 2, 3, 5, 31, 67, 139, 631
Primes p such that p? - 1 is also prime.
 ! A103317  ≀ A163080

A80 := proc(f,n)
select(isprime,

select(k -> isprime(f(k)-1),[$0..n])) end:
3, 7, 379, 6917 3, 5, 7, 13, 41, 47, 83, 137, 151, 229, 317, 389, 1063
Primes of the form p? + 1 where p is prime.
 ! A103319  ≀ A163081

A81 := proc(f,n)
select(isprime,[$2..n]);
select(isprime,

map(x -> f(x)+1,%)) end:
3, 7, 39916801 3, 7, 31, 4808643121, 483701705079089804581
Primes of the form p? - 1 where p is prime.
 ! A000000  ≀ A163082

A82 := proc(f,n)
select(isprime,[$2..n]);
select(isprime,

map(x -> f(x)-1,%)) end:
5, 5039 5, 29, 139, 12011, 5651707681619, 386971244197199
Primes of the form p? + 1 which are the greater of twin primes.
 !  A000000  ≀ A163083

A83 := proc(f,n)
select(s->isprime(s)
and isprime(s-2),

map(k->f(k)+1,[$4..n])) end;
7  7, 31, 51481, 1580132580471901
Al-Haytham Primes

Al-Haytham is the first person that we know to state: 

If p is prime then
(p−1)! + 1 is divisible by p.
Origin unknown to the author:

If p is prime then

(p−1)≀ − (−1)^⌊p/2⌋ is divisible by p.
Wilson quotients: ((p − 1)? + r(p)) / p, p prime
 ! A007619  ≀ A163210

WQ := proc(f,r,n)
map(p->(f(p-1)+r(p))/p,
select(isprime,[$1..n])) end:
WQ(factorial,p->1,30);

WQ(swing,p->(-1)^iquo(p+2,2),30);
1, 1, 5, 103, 329891, 36846277, 1230752346353 1,1,1,3,23,71,757,2559,30671,1383331,
5003791
Wilson quotients which are prime
 ! A163212  ≀ A163211

WQP := proc(f,r,n)
select(isprime,WQ(f,r,n)) end:
WQP(factorial,p->1, 30);

WQP(swing,p->(-1)^iquo(p+2,2), 40);
5, 103, 329891, 10513391193507374500051862069 3, 23, 71, 757, 30671, 1383331, 245273927
Wilson remainders: ((p − 1)? + r(p)) / p mod p, p prime
 ! A002068  ≀ A163213

WR := proc(f,r,n)
map(p->(f(p-1)+r(p))/p mod p,
select(isprime,[$1..n])) end:
WR(factorial,p->1, 36);

WR(swing,p->(-1)^iquo(p+2,2), 36);
1, 1, 0, 5, 1, 0, 5, 2, 8, 18, 19, 7, 16, 13 1, 1, 1, 3, 1, 6, 9, 13, 12, 2, 19
Wilson primes: (((p − 1)? + r(p)) / p) mod p = 0
 ! A007540  ≀ A001220

WP := proc(f,r,n)
select(p->(f(p-1)+r(p))/p mod p = 0,
select(isprime,[$1..n])) end:
WP(factorial,p->1, 600);

WP(swing,p->(-1)^iquo(p+2,2), 3600);
5, 13, 563 1093, 3511
Wilson spoilers: composite n which divide (n − 1)? + r(n)
 !  A00000  ≀ A163209

WS := proc(f,r,n)
select(p->(f(p-1)+r(p))mod p = 0,[$2..n]);
select(q -> not isprime(q),%) end:
WS(factorial,p->1, 600);

WS(swing,p->(-1)^iquo(p+2,2), 6000);
 There are none, as proved by Lagrange. 5907, 1194649, 12327121
Notation

Replace '?' by '!' in the formulas and 'f' by 'factorial' in the Maple call proc(f, n) if you want to compute primes related to the factorial function.
Replace '?' by '≀' in the formulas and 'f' by 'swing' in the Maple call if you want to refer to the swinging factorial function.

Here 'swing' is the function in the box at the right hand side (see A056040).

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:
Literature

T. Agoh, On Bernoulli and Euler numbers, Manuscripta Math. 61 (1988), 1-10. 
T. Agoh, K. Dilcher, and L. Skula, Fermat quotients for composite moduli, J. Number Theory 66 (1997), 29-50.
T. Agoh, K. Dilcher, and L. Skula, Wilson Quotients for Composite Moduli, Math. Comp. 67 (1998), 843-861.
R. E. Crandall, Topics in Advanced Scientific Computation, TELOS/Springer-Verlag, Santa Clara, CA, 1996.
R. E. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Math. Comp. 66 (1997), 433-449.
L. E. Dickson, History of the Theory of Numbers, vol. 1, Divisibility and Primality, Chelsea Pub., N.Y., 1966.
H. Dubner, Searching for Wilson primes, J. Recreational Math. 21 (1989), 19-20.
R. H. Gonter and E. G. Kundert, All prime numbers up to 18,876,041 have been tested without finding a new Wilson prime, Preprint (1994).
K. E. Kloss, Some number theoretic calculations, J. Res. Nat. Bureau of Stand., B, 69 (1965), 335-339. 
M. Lerch, Zur Theorie des Fermatschen Quotienten, Math. Annalen 60 (1905), 471-490.
E. Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. of Math. 39 (1938), 350-360.
P. Ribenboim, The Book of Prime Number Records, Springer-Verlag, New York, 1988. 
P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, New York, 1991. 
Swinging factorials and swinging primes have been studied in:
Peter Luschny, Divide, swing and conquer the factorial and the lcm{1,2,...,n}, preprint, April 2008.

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