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A240715 Primes p such that p*q*r + 6 and p*q*r - 6 are primes where q and r are the next two primes after p.

%I A240715 #11 Oct 29 2022 11:00:21
%S A240715 569,1531,1549,7103,7451,9013,10627,10853,11779,11783,12671,12941,
%T A240715 14821,14851,17489,18493,20717,20959,25237,26309,27739,29669,29873,
%U A240715 34549,35977,36251,37591,38351,38639,39551,40129,45589,46957,47317,48781,55163,55259
%N A240715 Primes p such that p*q*r + 6 and p*q*r - 6 are primes where q and r are the next two primes after p.
%H A240715 K. D. Bajpai, <a href="/A240715/b240715.txt">Table of n, a(n) for n = 1..1444</a>
%e A240715 569 is in the sequence because 569*571*577 + 6 = 187466729 and 569*571*577 - 6 = 187466717 are both prime where 571 and 577 are the next two primes after 569.
%e A240715 1531 is in the sequence because 1531*1543*1549 + 6 = 3659253823 and 1531*1543*1549 - 6 = 3659253811 are both prime where 1543 and 1549 are the next two primes after 1531.
%p A240715 KD := proc(n) local a,b,d; a:=ithprime(n)*ithprime(n+1)*ithprime(n+2); b:=a+6; d:=a-6; if  isprime(b) and isprime(d) then RETURN (ithprime(n)); fi; end: seq(KD(n), n=1..10000);
%t A240715 c = 0; Do[If[PrimeQ[Prime[n]*Prime[n+1]*Prime[n+2] +6] && PrimeQ[Prime[n]*Prime[n+1]*Prime[n+2] -6],c=c+1;Print[c, " ", Prime[n]]],{n,1,500000}];
%t A240715 KD={};   f=Prime[n+1]*Prime[n+2];  Do[p=Prime[n]; If[ PrimeQ[p*f+6] && PrimeQ[p*f-6], AppendTo[KD,p]], {n,10000}]; KD
%t A240715 Select[Partition[Prime[Range[6000]],3,1],AllTrue[Times@@#+{6,-6},PrimeQ]&][[All,1]] (* _Harvey P. Dale_, Oct 29 2022 *)
%o A240715 (Magma) [p: p in PrimesUpTo(10^5) | IsPrime(t-6) and IsPrime(t+6) where t is p*NextPrime(p)*NextPrime(NextPrime(p))]; // _Bruno Berselli_, Apr 11 2014
%Y A240715 Cf. A006094, A048880, A051507, A240596.
%K A240715 nonn
%O A240715 1,1
%A A240715 _K. D. Bajpai_, Apr 10 2014