A339920 -id:A339920 - cp's OEIS frontend

A339698 Primes of the form p^2 - p*q + q^2, where p and q are consecutive primes.

7, 19, 3163, 23743, 28927, 70783, 141403, 198943, 223837, 265333, 283267, 329503, 1136383, 1223263, 1254427, 1488427, 2238043, 2421163, 3625243, 3904603, 4709143, 4884127, 5216683, 5784133, 7376683, 8065627, 8797183, 10660333, 11242717, 12348223, 16613803, 18594019, 19202167, 19999027

Offset: 1

Zak Seidov, Dec 13 2020

nonn

			a(2) = 19 = 3^2 - 3*5 + 5^2 is the only prime obtained with a pair of twin primes: (3, 5). - _Bernard Schott_, Dec 23 2020

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 100 terms from Zak Seidov)

Cf. A243761 (similar, with p^2 + p*q + q^2), A339920 (the primes p).

Select[Map[#1^2 - #1 #2 + #2^2 & @@ # &, Partition[Prime@ Range[610], 2, 1]], PrimeQ] (* Michael De Vlieger, Dec 13 2020 *)

forprime(p=1, 1e4, my(q=nextprime(p+1), x=p^2-p*q+q^2); if(ispseudoprime(x), print1(x, ", "))) \\ Felix Fröhlich, Dec 14 2020

first(n) = { my(q = 2, p, res = vector(n), t = 0); forprime(p = 3, oo, c = p^2 - p*q + q^2; if(isprime(c), t++; res[t] = c; if(t >= n, return(res) ) ); q = p; ) } \\ David A. Corneth, Dec 19 2020

A342705 Primes p such that (p^2 - p*q + q^2)/3 is prime, where q is the next prime after p.

Original entry on oeis.org

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

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

nonn

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

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

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

Cf. A339920, A342706.

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;

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

cp's OEIS Frontend

A339698 Primes of the form p^2 - p*q + q^2, where p and q are consecutive primes.

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