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A327741 Terms of A002496 that are the average of two distinct terms of A002496. 1
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 (list; graph; refs; listen; history; text; internal format)
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

od:

R;

CROSSREFS

Cf. A002496.

Sequence in context: A082435 A136098 A153807 * A183056 A261881 A262633

Adjacent sequences:  A327738 A327739 A327740 * A327742 A327743 A327744

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified October 26 08:00 EDT 2021. Contains 348267 sequences. (Running on oeis4.)