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)^floor(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, or 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). A Maple worksheet is here. 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. | 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: |
|
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