A327378 - OEIS

%I #28 Oct 04 2019 04:59:00

%S 863,1303,2539,2953,3251,3457,4007,4139,4507,5209,5431,5717,7229,7867,

%T 7933,9323,9421,11821,12011,12101,12143,12907,12983,13709,13859,14071,

%U 14549,15661,16141,16811,17977,18773,18899,19069,19577,20347,20533,21013,21503,21599,22543

%N Smallest prime p in a sequence of six consecutive primes (p,q,r,u,v,w) for which the conic section discriminant Delta < 0 for the general conic section px^2 + qy^2 + rz^2 + 2uyz + 2vxz + 2wxy = 0.

%C Delta = pqr + 2uvw - pu^2 - qv^2 - rw^2.

%C Using consecutive primes in the general conic section and computing Delta, the value is most often (~98%) > 0.

%e For (p,q,r,u,v,w) = (2,3,5,7,11,13), Delta = 726 > 0. Hence, p=2 (smallest prime) is not in the sequence.

%e For (p,q,r,u,v,w) = (863,877,881,883,887,907), Delta = -73164 < 0. Hence, p=863 (smallest prime) is a member of the sequence.

%t Select[Partition[Prime@ Range[3000], 6, 1], Function[{p, q, r, u, v, w}, p q r + 2 u v w - p u^2 - q v^2 - r w^2 < 0] @@ # &][[All, 1]] (* _Michael De Vlieger_, Sep 30 2019 *)

%o (PARI) lista(nn) = {forprime (p=1, nn, q = nextprime(p+1); r = nextprime(q+1); u = nextprime(r+1); v = nextprime(u+1); w = nextprime(v+1); if ((x=p*q*r + 2*u*v*w - p*u^2 - q*v^2 - r*w^2)< 0, print1(p, ", ")););} \\ _Michel Marcus_, Sep 18 2019

%Y Cf. A000040.

%K nonn

%O 1,1

%A _Philip Mizzi_, Sep 16 2019

%E More terms from _Michel Marcus_, Sep 18 2019