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A232768 Numbers n with the property that n^2+(n+1)^2 and n^2+(n+1)^2+(n+2)^2 are both prime. 0
2, 12, 14, 24, 34, 122, 154, 164, 272, 342, 464, 612, 674, 734, 784, 794, 854, 1174, 1262, 1274, 1364, 1392, 1524, 1554, 1664, 1682, 1844, 1854, 1862, 1892, 1924, 1942, 1994, 2232, 2294, 2354, 2442, 2592, 2802, 2884, 3124, 3164, 3292, 3394, 3544, 3594, 3632, 3724, 3892, 3904, 3922 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

See A027862 for primes of the form x^2+(x+1)^2 = 2x^2+2x+1.

See A027864 for primes of the form x^2+(x+1)^2+(x+2)^2 = 3x^2+6x+5.

It is an open question whether either of these polynomials produces an infinite number of primes.  This sequence lists the values of x that produce a prime in both polynomials. x must be congruent to 0 or 2 (mod 4) and all the generated primes are of the form 4k+1.

REFERENCES

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 2005, page 266.

LINKS

Table of n, a(n) for n=1..51.

Wikipedia, Hardy and Littlewood's Conjecture F.

EXAMPLE

When x=14, 2x^2+2x+1=421 and 3x^2+6x+5=677. 14 is the third value of x for which both these polynomials produce a prime number, so a(3)=14.

MATHEMATICA

lst = {}; Do[If[And[PrimeQ[n^2 + (n + 1)^2], PrimeQ[n^2 + (n + 1)^2 + (n + 2)^2]], Print[n]; AppendTo[lst, n]], {n, 10000}]

CROSSREFS

Cf. A027862, A027864. Equals n common to A027861 and A027863.

Sequence in context: A124163 A108976 A073598 * A022368 A076484 A259128

Adjacent sequences:  A232765 A232766 A232767 * A232769 A232770 A232771

KEYWORD

nonn,easy

AUTHOR

Chris Fry, Nov 29 2013

STATUS

approved

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Last modified March 19 11:10 EDT 2019. Contains 321329 sequences. (Running on oeis4.)