A327741 Terms of A002496 that are the average of two distinct terms of A002496.

101, 21317, 24337, 462401, 1073297, 1123601, 1263377, 1887877, 1943237, 2446097, 2604997, 2890001, 3422501, 4202501, 4343057, 5354597, 6330257, 7862417, 8386817, 8410001, 9156677, 10536517, 10719077, 11383877, 12068677, 12110401, 12503297, 16273157, 18062501, 19219457, 21771557, 22429697

Offset: 1

Author

J. M. Bergot and Robert Israel, Sep 23 2019

Keywords

• nonn

Comments

Primes of the form x^2+1 such that 2*x^2=y^2+z^2 where y^2+1 and z^2+1 are primes.

Some terms of the sequence are the average of more than one pair of terms of A002496. E.g., 2890001 = (115601 + 5664401)/2 = (2016401 + 3763601)/2, while 5354597 = (42437 + 10666757)/2 = (1136357 + 9572837)/2 = (1552517 + 9156677)/2.

Primes of the form u^2*(s^2 + t^2)^2 + 1 where u^2*(s^2 + 2*s*t - t^2)^2 + 1 and u^2*(-s^2 + 2*s*t + t^2)^2 + 1 are prime, (sqrt(2) - 1)*s < t < s. The generalized Bunyakovsky conjecture implies there are infinitely many terms for each such pair (s,t).

Examples

			a(3)=24337 is in the sequence because 24337=(7057+41617)/2 with 7057, 24337 and 41617 all terms of A002496, i.e., they are primes and 7057=84^2+1, 24337=156^2+1 and 41617=204^2+1.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..1055

Crossrefs

Cf. A002496.

Programs

Maple

N:= 10^8: # to get terms <= N
P:= select(isprime, [seq(x^2+1, x=2..floor(sqrt(N-1)),2)]):
nP:= nops(P):
R:= NULL:
for i from 1 to nP do
  x:= P[i];
  for j from 1 to i-1 do
    z:= 2*x-P[j];
    if issqr(z-1) and isprime(z) then R:= R, x; break fi
  od
od:
R;