A340136 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.

23, 43, 53, 73, 83, 101, 127, 131, 139, 151, 157, 179, 181, 229, 281, 283, 293, 307, 311, 313, 349, 353, 359, 389, 397, 401, 409, 419, 443, 449, 457, 461, 463, 487, 491, 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

Offset

1,1

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

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.

Maple

p2:=2: p3:= 3: p4:= 5: count:= 0: R:= NULL:
while count < 100 do
  p1:= p2; p2:= p3; p3:= p4; p4:= nextprime(p4);
  w:= p1*p2+p3*p4;
  if not ormap(t -> isprime(t+w), [p1, p2, p3, p4]) then
    count:= count+1; R:= R, p1
  fi
od:
R;

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A340126.

Sequence in context: A331342 A306085 A037137 * A154530 A156979 A281226

Adjacent sequences: A340133 A340134 A340135 * A340137 A340138 A340139

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Dec 29 2020

Status

approved