A327592 Smallest prime (p) of six consecutive primes (p,q,r,u,v,w) for which the conic section discriminant (Delta) is a perfect square.

397, 68219, 87881, 316531, 430487, 440653, 639701, 691813, 732497, 982981, 1145773, 1226683, 1288337, 1291223, 1537751, 1563943, 1756663, 1913803, 2043397, 2134589, 2143391, 2317097, 2366789, 2528833, 3047311, 3107597, 3261523, 3678869, 3884389, 4143397

Offset

1,1

Comments

Delta = pqr + 2uvw - pu^2 - qv^2 - rw^2 for the general conic section px^2 + qy^2 + rz^2 + 2uyz + 2vxz + 2wxy = 0.

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.)

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 192 terms from Philip Mizzi)

Example

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.

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A000040.

Sequence in context: A338476 A152309 A298124 * A340419 A191942 A158316

Adjacent sequences: A327589 A327590 A327591 * A327593 A327594 A327595

Keyword

nonn

Author

Philip Mizzi, Sep 18 2019

Status

approved