login
A342705
Primes p such that (p^2 - p*q + q^2)/3 is prime, where q is the next prime after p.
2
5, 7, 13, 17, 19, 59, 97, 101, 107, 109, 191, 223, 229, 277, 283, 569, 613, 631, 643, 709, 719, 743, 829, 857, 881, 1031, 1049, 1051, 1091, 1109, 1171, 1193, 1249, 1277, 1301, 1327, 1489, 1579, 1637, 1697, 1949, 1979, 2003, 2081, 2089, 2113, 2141, 2203, 2357, 2423, 2539, 2593, 2659, 2749, 2789, 2819
OFFSET
1,1
COMMENTS
If p == -q (mod 3) then p^2 - p*q + q^2 is divisible by 3.
LINKS
EXAMPLE
a(5) = 19 is a term because 19 and 23 are consecutive primes and (19^2 - 19*23 + 23^2)/3 = 151 is prime.
MAPLE
R:= NULL: q:= 2: count:= 0:
while count < 100 do
p:= q; q:= nextprime(p);
r:= (p^2-p*q+q^2)/3;
if r::integer and isprime(r) then
count:= count+1; R:= R, p;
fi;
od:
R;
PROG
(Python)
from sympy import isprime, nextprime
def aupto(limit):
p, q, num, alst = 2, 3, 7, []
while p <= limit:
if num%3 == 0 and isprime(num//3): alst.append(p)
p, q = q, nextprime(q)
num = p**2 - p*q + q**2
return alst
print(aupto(2819)) # Michael S. Branicky, Mar 18 2021
CROSSREFS
Sequence in context: A106067 A287614 A320866 * A314321 A314322 A028311
KEYWORD
nonn
AUTHOR
J. M. Bergot and Robert Israel, Mar 18 2021
STATUS
approved