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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

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

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

Cf. A339920, A342706.

Sequence in context: A106067 A287614 A320866 * A314321 A314322 A028311

Adjacent sequences:  A342702 A342703 A342704 * A342706 A342707 A342708

KEYWORD

nonn

AUTHOR

J. M. Bergot and Robert Israel, Mar 18 2021

STATUS

approved

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Last modified January 21 20:28 EST 2022. Contains 350480 sequences. (Running on oeis4.)