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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 27 06:10 EST 2021. Contains 349363 sequences. (Running on oeis4.)