OFFSET
1,1
COMMENTS
This sequence provides a complete listing of primes of the form k!-p ordered by groups that are adjacent to each other respectively to intervals [k!/2, k!], for k=3,4,5,..., till there are no further primes in the current interval.
This demonstrates that most of the primes in the interval [k!/2, k!], for k>4, have the form k!-p and are symmetric about the half of a factorial number with initial primes p, but not all of them.
Conjecture 1: every prime number in the interval [k!-k^2, k!-k] for k>4 has only the form k!-p where p is the n-th prime.
It is easily proved that if q is the prime in the interval [k!-k^2, k!-k] for k>4, then k!-q is always a prime number.
LINKS
EXAMPLE
The first terms 3=3!-3 and 13=4!-11, 17=4!-7, 19=4!-5 are primes of the form k!-p in the intervals [3!/2, 3!] and [4!/2, 4!] accordingly.
Each of the next terms from the interval [5!/2, 5!]: 61, 67, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113 has the form 5!-p where p = 59, 53, 47, 41, 37, 31, 23, 19, 17, 13, 11, 7 respectively.
MATHEMATICA
NN = 8; list = {}; For[k = NN, k >= 2, k--, i = 1; While[(p = Prime[i++]) <= k!/2, If[PrimeQ[k! - p], AppendTo[list, k! - p]; ]; ]; ]; list2=Sort[list]; Print[list2];
PROG
(PARI) lista(NN) = for(n=3, NN, forprime(p=2, n!\2, if(isprime(n!-p), print1(n!-p, ", ")))) \\ Jinyuan Wang, Jul 22 2019
CROSSREFS
KEYWORD
easy,nonn,tabf
AUTHOR
Vasily Bashilov, Jul 21 2019
EXTENSIONS
More terms from Jinyuan Wang, Jul 22 2019
STATUS
approved