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%I #29 Oct 18 2019 01:03:57
%S 397,68219,87881,316531,430487,440653,639701,691813,732497,982981,
%T 1145773,1226683,1288337,1291223,1537751,1563943,1756663,1913803,
%U 2043397,2134589,2143391,2317097,2366789,2528833,3047311,3107597,3261523,3678869,3884389,4143397
%N Smallest prime (p) of six consecutive primes (p,q,r,u,v,w) for which the conic section discriminant (Delta) is a perfect square.
%C Delta = pqr + 2uvw - pu^2 - qv^2 - rw^2 for the general conic section px^2 + qy^2 + rz^2 + 2uyz + 2vxz + 2wxy = 0.
%C Perfect squares of this form are quite rare, representing approximately 0.0048% of possible Delta values using consecutive prime number coefficients. (First 4 million primes tested.)
%H Giovanni Resta, <a href="/A327592/b327592.txt">Table of n, a(n) for n = 1..10000</a> (first 192 terms from Philip Mizzi)
%e 48 = sqrt(2304) = pqr + 2uvw - pu^2 - qv^2 - rw^2 for (p,q,r,u,v,w) = (440653,440669,440677,440681,440683,440711), which are consecutive primes. Hence, 440653 is a member of the sequence.
%t f[{p_, q_, r_, u_, v_, w_}] := p q r + 2 u v w - p u^2 - q v^2 - r w^2; First /@ Select[Partition[ Prime@ Range@ 300000, 6, 1], IntegerQ@ Sqrt@ f@ # &] (* _Giovanni Resta_, Sep 30 2019 *)
%o (PARI) chk(nn) = {forprime (p=1, nn, my(q = nextprime(p+1), r = nextprime(q+1), u = nextprime(r+1), v = nextprime(u+1), w = nextprime(v+1)); if (issquare(p*q*r + 2*u*v*w - p*u^2 - q*v^2 - r*w^2), print1(p, ", ")););} \\ _Michel Marcus_, Sep 30 2019
%Y Cf. A000040.
%K nonn
%O 1,1
%A _Philip Mizzi_, Sep 18 2019
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