A327741 - OEIS

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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

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OFFSET

1,1

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).

LINKS

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

EXAMPLE

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.

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:

CROSSREFS