A340037 Primes p such that (p^2+q^2)/2 and (q^2 + 2*p*q - p^2)/2 are prime, where q is the next prime after p.

3, 5, 13, 199, 419, 421, 1187, 1319, 1693, 1783, 2029, 2069, 2861, 3041, 3559, 3881, 4493, 4523, 4957, 5153, 6359, 7187, 7193, 8171, 8293, 8623, 8719, 8783, 9629, 10631, 12601, 13829, 13831, 15013, 15817, 16183, 16339, 17519, 18169, 18593, 18773, 18913, 19219, 19301, 19379, 19597, 20201, 20533

Offset

1,1

Comments

p*q, (q^2-p^2)/2 and (p^2+q^2)/2 are the sides of a right triangle whose hypotenuse and sum of the other two sides are prime.

Links

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

Example

a(3) = 13 is a term because it is prime, the next prime is 17, and (13^2+17^2)/2 = 229 and (17^2+2*13*17-13^2)/2 = 281 are prime.

Maple

R:= NULL: count:= 0: q:= 3:
while count < 100 do
  p:= q; q:= nextprime(q);
  if isprime((p^2+q^2)/2) and isprime((q^2+2*p*q-p^2)/2) then
   count:= count+1; R:= R, p;
  fi
od:
R;

Prog

(PARI) isok(p) = if ((p>2) && isprime(p), my(q=nextprime(p+1)); isprime((p^2+q^2)/2) && isprime((q^2 + 2*p*q - p^2)/2)) \\ Michel Marcus, Dec 27 2020

Crossrefs

Sequence in context: A087170 A123370 A110407 * A062698 A200769 A128341

Adjacent sequences: A340034 A340035 A340036 * A340038 A340039 A340040

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Dec 26 2020

Status

approved