A348353 Primes p such that p^2+q+r, p+q^2+r and p+q+r^2 are all prime, where q and r are the next two primes after p.

11, 379, 419, 641, 7433, 10177, 32609, 33809, 37511, 41081, 54851, 58313, 63671, 68491, 71069, 85427, 106163, 112087, 116741, 117167, 127373, 128489, 137483, 155689, 157793, 180053, 197003, 208577, 209327, 210659, 213467, 227501, 234599, 248099, 276439, 278279, 288461, 298667, 313109, 317701, 325421

Offset

1,1

Comments

The generalized Bunyakovsky conjecture implies that there are infinitely many terms, in fact infinitely many for which q = p+2 and r = p+6. - Robert Israel, Oct 21 2021

Links

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

Example

a(1) = 11 is a term because with p=11, q=13, r=17 we have p^2+q+r = 151, p+q^2+r = 197, p+q+r^2 = 313, all of which are primes.

Maple

R:= NULL: count:= 0:
p:= 2: q:= 3: r:= 5:
while count < 50 do
  p:= q; q:= r; r:= nextprime(r);
  if andmap(isprime, [p^2+q+r, p+q^2+r, p+q+r^2]) then
    count:= count+1; R:= R, p;
  fi
od:
R;

Mathematica

Select[Partition[Select[Range[350000], PrimeQ], 3, 1], And @@ PrimeQ[{#[[1]]^2 + #[[2]] + #[[3]], #[[1]] + #[[2]]^2 + #[[3]], #[[1]] + #[[2]] + #[[3]]^2}] &][[;; , 1]] (* Amiram Eldar, Oct 14 2021 *)

Prog

(Python)
from itertools import islice
from sympy import nextprime, isprime
def A348353(): # generator of terms.
    p, q, r = 2, 3, 5
    while True:
        if isprime(p*p+q+r) and isprime(p+q*q+r) and isprime(p+q+r*r):
            yield p
        p, q, r = q, r, nextprime(r)
A348353_list = list(islice(A348353(), 20)) # Chai Wah Wu, Oct 21 2021

Crossrefs

Sequence in context: A018893 A051862 A006698 * A048431 A286914 A354513

Adjacent sequences: A348350 A348351 A348352 * A348354 A348355 A348356

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Oct 14 2021

Status

approved