A243761 Primes of the form p^2 + pq + q^2, where p and q are consecutive primes.

19, 109, 433, 1327, 4567, 6079, 19687, 49927, 62233, 103813, 160087, 172801, 238573, 363313, 395323, 463363, 583447, 640333, 753007, 1145773, 1529413, 1728247, 1968301, 2056753, 2223967, 2317927, 2349679, 2413927, 3121201, 3577393, 4148953, 4298443

Offset: 1

Author

K. D. Bajpai, Jun 10 2014

Keywords

• nonn

Examples

			19 is in the sequence because 2^2 + 2*3 + 3^2 = 19 is prime: 2 and 3 are consecutive primes.
109 is in the sequence because 5^2 + 5*7 + 7^2 = 109 is prime: 5 and 7 are consecutive primes.
		

Links

• K. D. Bajpai, Table of n, a(n) for n = 1..8900

Crossrefs

Cf. A000040, A007645, A003136.

Programs

Maple

with(numtheory): A243761:= proc() local k, p, q; p:=ithprime(n); q:=ithprime(n+1); k:=p^2 + p*q + q^2;  if isprime(k) then RETURN (k); fi; end: seq(A243761 (), n=1..500);

Mathematica

Select[Table[Prime[n]^2 + Prime[n] Prime[n + 1] + Prime[n + 1]^2, {n, 500}], PrimeQ[#] &]

Python

from itertools import islice
from sympy import isprime, nextprime
def A243761_gen(): # generator of terms
    p, q = 2, 3
    while True:
        if isprime(r:=p*(p+q)+q**2):
            yield r
        p, q = q, nextprime(q)
A243761_list = list(islice(A243761_gen(),20)) # Chai Wah Wu, Feb 27 2023