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User:Peter Luschny/SwingingPrimes
Swinging Primes
Factorial primes, primes which are within 1 of a factorial number. |
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! A088054 | ≀ A163074 |
A74 := proc(f,n) |
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) |
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) |
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) |
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) |
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) |
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) |
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) |
3, 7, 39916801 | 3, 7, 31, 4808643121, 483701705079089804581 | |
Primes of the form p? - 1 where p is prime. | ||
! A000000 | ≀ A163082 |
A82 := proc(f,n) |
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) |
7 | 7, 31, 51481, 1580132580471901 | |
Al-Haytham Primes | ||
Al-Haytham is the first person that we know to state: (p−1)! + 1 is divisible by p. |
Origin unknown to the author: If p is prime then |
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Wilson quotients: ((p − 1)? + r(p)) / p, p prime | ||
! A007619 | ≀ A163210 |
WQ := proc(f,r,n) |
1, 1, 5, 103, 329891, 36846277, 1230752346353 | 1,1,1,3,23,71,757,2559,30671,1383331, 5003791 |
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Wilson quotients which are prime | ||
! A163212 | ≀ A163211 |
WQP := proc(f,r,n) |
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) |
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) |
5, 13, 563 | 1093, 3511 | |
Wilson spoilers: composite n which divide (n − 1)? + r(n) | ||
! A00000 | ≀ A163209 |
WS := proc(f,r,n) |
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. |
swing := proc(n) |
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Literature | ||
T. Agoh, On Bernoulli and Euler numbers, Manuscripta Math. 61 (1988), 1-10. |