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A309291
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Primes of the form k!-p where p <= k!/2 is also a prime and k=3,4,5,....
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1
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3, 13, 17, 19, 61, 67, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 367, 373, 383, 389, 409, 439, 443, 449, 457, 463, 479, 487, 491, 509, 521, 523, 541, 547, 557, 563, 569, 571, 593, 607, 613, 617, 619, 631, 641, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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];
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PROG
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(PARI) lista(NN) = for(n=3, NN, forprime(p=2, n!\2, if(isprime(n!-p), print1(n!-p, ", ")))) \\ Jinyuan Wang, Jul 22 2019
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CROSSREFS
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KEYWORD
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easy,nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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