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%I #27 Feb 03 2019 17:01:45
%S 5,41,47,71,83,101,107,113,733,739,757,827,829,859,863,883,887,947,
%T 953,977,983,997,1019,1039,1069,1093,1097,1103,1109,1123,1129,1187,
%U 1193,1223,1229,1237,1283,1289,1297,1303,1307,1327,1367,1373,1429,1433,1439,1453,1459
%N Primes p that are the sum of another prime q and the greatest k such that k! < p.
%C Primes p such that p - A084558(p) is also a prime.
%C Inspired by the fact that this sequence has a nonempty intersection with A124598.
%C Let H be the intersection of this sequence and A124598. With some observed exceptions q = 3, 103, 1013, 1223, 2903, ..., if some p in this sequence is in H then the corresponding q also belongs to H. It is unknown whether H is finite.
%C Based upon the first terms of this sequence, it looks like those k satisfying the definition given in NAME are the even terms of A084558 repeated in certain weakly increasing sequence.
%H Robert Israel, <a href="/A302692/b302692.txt">Table of n, a(n) for n = 1..10000</a>
%e a(3) = 41 since 41 is the third prime that satisfies the given definition(s) for this sequence. Note that A084558(41) = 4; if we subtract: 41-4 = 37, which is also a prime. Also p = 41 and q = 37 are consecutive terms in A124598.
%e a(300) = 49459, because q = 49459 - A084558(49459) = 49451 is a prime and p = 49459 is the 300th prime satisfying such property. Also p and q are consecutive terms in A124598.
%p k:= 0: f:= 1: p:= 1: Res:= NULL: count:= 0:
%p while count < 100 do
%p p:= nextprime(p);
%p while p > f do k:= k+1; f:= f*(k+1) od;
%p if isprime(p-k) then Res:= Res, p; count:= count+1 fi;
%p od:
%p Res; # _Robert Israel_, Jun 10 2018
%t Select[Prime@ Range[2^8], PrimeQ[# - Block[{k = 1}, While[k! <= #, k++]; k - 1]] &] (* _Michael De Vlieger_, Apr 10 2018 *)
%o (PARI) A084558(n)={my(m=0);while(n\=m++,);m-1}
%o firstTerms(U)={my(L:list=List());forprime(p=2,nextprime(U),if(ispseudoprime(p-A084558(p)),listput(L,p)));return(Vec(L))}
%Y Cf. A084558, A124598.
%K nonn,look
%O 1,1
%A _R. J. Cano_, Apr 08 2018
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