%I #44 Dec 01 2023 13:31:46

%S 59,73,79,89,101,109,197,211,239,241,263,281,307,337,367,373,379,409,

%T 419,421,439,443,449,461,463,491,523,547,557,571,593,601,613,617,631,

%U 647,653,659,673,701,709,769,797,811,839,853,863,881,907,929,937,941,967

%N Primes p such that p - m! is composite, where m is the greatest number such that m! < p.

%C The first five terms 59, 73, 79, 89, 101 belong to A023209. The terms 409, 419, 421, 439, 443, 449 also belong to A127209.

%C It seems likely that a(n) ~ n log n, can this be proved? - _Charles R Greathouse IV_, Sep 20 2012

%H Charles R Greathouse IV, <a href="/A212266/b212266.txt">Table of n, a(n) for n = 1..10000</a>

%e 29 is not a member because 29 - 4! = 5 is prime.

%e 59 is a member because 59 - 4! = 35 is composite.

%t Select[Prime[Range[200]],Module[{m=9},CompositeQ[While[m!>=#,m--];#-m!]]&] (* The initial m constant (set at 9 in the program) needs to be increased if the prime Range constant (set at 200 in the program) is increased beyond 30969. *) (* _Harvey P. Dale_, Dec 01 2023 *)

%o (PARI) for(n=3,5,N=n!;forprime(p=N+3,N*(n+1),if(!isprime(p-N), print1(p", ")))) \\ _Charles R Greathouse IV_, May 12 2012

%o (PARI) is_A212266(p)=isprime(p) && for(n=1,p, n!<p || return(bigomega(p-(n-1)!)>1)) \\ _M. F. Hasler_, May 20 2012

%Y Cf. A212600, A212598, A136437, A023209, A127209.

%K nonn,easy

%O 1,1

%A _Balarka Sen_, May 12 2012