A340136 - OEIS

%I #11 Jul 05 2021 19:46:58

%S 23,43,53,73,83,101,127,131,139,151,157,179,181,229,281,283,293,307,

%T 311,313,349,353,359,389,397,401,409,419,443,449,457,461,463,487,491,

%U 509,521,563,569,571,601,617,631,641,643,653,673,709,739,757,769,787,797,809,827,829,839,857,863,877

%N Primes p1 such that, with p2, p3, p4 the next three primes, p1*p2+p3*p4+p1, p1*p2+p3*p4+p2, p1*p2+p3*p4+p3 and p1*p2+p3*p4+p4 are all composite.

%H Robert Israel, <a href="/A340136/b340136.txt">Table of n, a(n) for n = 1..10000</a>

%e a(3) = 53 is a term because with p1 = 53, p2 = 59, p3 = 61, p4 = 67, we have p1*p2+p3*p4+p1 = 7267, p1*p2+p3*p4+p2 = 7273, p1*p2+p3*p4+p3 = 7275, p1*p2+p3*p4+p4 = 7281, all composite.

%p p2:=2: p3:= 3: p4:= 5: count:= 0: R:= NULL:

%p while count < 100 do

%p   p1:= p2; p2:= p3; p3:= p4; p4:= nextprime(p4);

%p   w:= p1*p2+p3*p4;

%p   if not ormap(t -> isprime(t+w), [p1, p2, p3, p4]) then

%p     count:= count+1; R:= R, p1

%p   fi

%p od:

%p R;

%t acQ[{a_,b_,c_,d_}]:=AllTrue[a*b+c*d+{a,b,c,d},CompositeQ]; Select[ Partition[ Prime[ Range[200]],4,1],acQ][[All,1]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jul 05 2021 *)

%o (PARI) isA340136(p1) = if(!isprime(p1),0,my(p2=nextprime(1+p1),p3=nextprime(1+p2),p4=nextprime(1+p3),x=((p1*p2)+(p3*p4))); !isprime(x+p1)&&!isprime(x+p2)&&!isprime(x+p3)&&!isprime(x+p4)); \\ _Antti Karttunen_, Dec 29 2020

%Y Cf. A340126.

%K nonn

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Dec 29 2020