login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Primes p such that (p^2 - p*q + q^2)/3 is prime, where q is the next prime after p.
2

%I #12 Mar 23 2021 05:40:54

%S 5,7,13,17,19,59,97,101,107,109,191,223,229,277,283,569,613,631,643,

%T 709,719,743,829,857,881,1031,1049,1051,1091,1109,1171,1193,1249,1277,

%U 1301,1327,1489,1579,1637,1697,1949,1979,2003,2081,2089,2113,2141,2203,2357,2423,2539,2593,2659,2749,2789,2819

%N Primes p such that (p^2 - p*q + q^2)/3 is prime, where q is the next prime after p.

%C If p == -q (mod 3) then p^2 - p*q + q^2 is divisible by 3.

%H Robert Israel, <a href="/A342705/b342705.txt">Table of n, a(n) for n = 1..10000</a>

%e a(5) = 19 is a term because 19 and 23 are consecutive primes and (19^2 - 19*23 + 23^2)/3 = 151 is prime.

%p R:= NULL: q:= 2: count:= 0:

%p while count < 100 do

%p p:= q; q:= nextprime(p);

%p r:= (p^2-p*q+q^2)/3;

%p if r::integer and isprime(r) then

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

%p fi;

%p od:

%p R;

%o (Python)

%o from sympy import isprime, nextprime

%o def aupto(limit):

%o p, q, num, alst = 2, 3, 7, []

%o while p <= limit:

%o if num%3 == 0 and isprime(num//3): alst.append(p)

%o p, q = q, nextprime(q)

%o num = p**2 - p*q + q**2

%o return alst

%o print(aupto(2819)) # _Michael S. Branicky_, Mar 18 2021

%Y Cf. A339920, A342706.

%K nonn

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Mar 18 2021