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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.

%I A232768 #17 Jul 30 2023 10:04:11
%S A232768 2,12,14,24,34,122,154,164,272,342,464,612,674,734,784,794,854,1174,
%T A232768 1262,1274,1364,1392,1524,1554,1664,1682,1844,1854,1862,1892,1924,
%U A232768 1942,1994,2232,2294,2354,2442,2592,2802,2884,3124,3164,3292,3394,3544,3594,3632,3724,3892,3904,3922
%N A232768 Numbers n with the property that n^2+(n+1)^2 and n^2+(n+1)^2+(n+2)^2 are both prime.
%C A232768 See A027862 for primes of the form x^2+(x+1)^2 = 2x^2+2x+1.
%C A232768 See A027864 for primes of the form x^2+(x+1)^2+(x+2)^2 = 3x^2+6x+5.
%C A232768 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.
%D A232768 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 2005, page 266.
%H A232768 Harvey P. Dale, <a href="/A232768/b232768.txt">Table of n, a(n) for n = 1..1000</a>
%H A232768 Wikipedia, <a href="http://en.wikipedia.org/wiki/Ulam_spiral#Hardy_and_Littlewood.27s_Conjecture_F">Hardy and Littlewood's Conjecture F</a>.
%e A232768 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.
%t A232768 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}]
%t A232768 Select[Range[2,4000,2],AllTrue[{(#^2+(#+1)^2),(#^2+(#+1)^2+(#+2)^2)},PrimeQ]&] (* _Harvey P. Dale_, Jul 30 2023 *)
%Y A232768 Cf. A027862, A027864. Equals n common to A027861 and A027863.
%K A232768 nonn,easy
%O A232768 1,1
%A A232768 _Chris Fry_, Nov 29 2013