A343448 Primes p such that p+q*(r+s) is prime, where p,q,r,s are consecutive primes.

5, 7, 11, 13, 19, 29, 37, 67, 71, 101, 103, 107, 193, 223, 229, 281, 293, 337, 359, 367, 541, 569, 613, 631, 647, 677, 709, 751, 809, 823, 829, 857, 881, 887, 919, 947, 971, 1009, 1019, 1049, 1237, 1249, 1279, 1373, 1439, 1471, 1543, 1571, 1627, 1637, 1693, 1733, 1783, 1907, 1993, 2017, 2161

Offset

1,1

Comments

Includes p if p, q = p+2, r = p+6, s = p+8 are consecutive primes and 2*p^2+19*p+28 is prime. The generalized Dickson's conjecture implies there are infinitely many such p.

Links

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

Example

a(3) = 11 is a term because 11,13,17 and 19 are consecutive primes and 11+13*(17+19) = 479 is prime.

Maple

A:= NULL: q:= 2: r:= 3: s:= 5: count:= 0:
while count < 100 do
  p:= q; q:= r; r:= s; s:= nextprime(s);
  v:= p+q*(r+s);
  if isprime(v) then A:= A, p; count:= count+1 fi
od:
A;

Prog

(Python)
from sympy import isprime, nextprime
def aupto(limit):
  p, q, r, s, alst = 2, 3, 5, 7, []
  while p <= limit:
    if isprime(p + q*(r+s)): alst.append(p)
    p, q, r, s = q, r, s, nextprime(s)
  return alst
print(aupto(2161)) # Michael S. Branicky, Apr 15 2021

Crossrefs

Cf. A343449.

Sequence in context: A267945 A088664 A023219 * A045438 A176579 A154275

Adjacent sequences: A343445 A343446 A343447 * A343449 A343450 A343451

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Apr 15 2021

Status

approved