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Primes of the form n^2 + 4n + 8.
3

%I #16 Sep 08 2022 08:45:14

%S 13,29,53,173,229,293,733,1093,1229,1373,2029,2213,3253,4229,4493,

%T 5333,7229,7573,9029,9413,10613,13229,13693,15629,18229,18773,21613,

%U 24029,26573,27893,31333,33493,37253,41213,42853,46229,47093,54293,55229

%N Primes of the form n^2 + 4n + 8.

%C Or, primes that are equal to the mean of 7 consecutive squares. - _Zak Seidov_, Apr 14 2007

%C Sum of 7 consecutive squares starting with m^2 is equal to 7*(13 + 6*m + m^2) and mean is (13 + 6*m + m^2)=(m+3)^2+4. Hence a(n)=A005473(n+1). Note that only nonnegative m's are considered. - _Zak Seidov_, Apr 14 2007

%C a(n)==1 (mod 4).

%C a(n)= A005473(n+1). - _Zak Seidov_, Apr 12 2007

%H Vincenzo Librandi, <a href="/A098062/b098062.txt">Table of n, a(n) for n = 1..1000</a>

%e 13 = (0^2 + ... + 6^2)/7, 29 = (2^2 + ... + 8^2)/7 = 29, 53 = (4^2 + ... + 10^2)/7 = 53.

%t Select[ Table[ n^2 + 4n + 8, {n, 240}], PrimeQ[ # ] &] (* _Robert G. Wilson v_, Sep 14 2004 *)

%o (PARI) for(n=0,240,if(isprime(p=n^2+4*n+8),print1(p,","))) \\ _Klaus Brockhaus_

%o (Magma) [a: n in [0..250] | IsPrime(a) where a is n^2 + 4*n + 8]; // _Vincenzo Librandi_, Jul 17 2012

%Y Cf. A005473, A056899, A067201, A007591, A129389, A129412.

%K nonn,easy

%O 1,1

%A _Giovanni Teofilatto_, Sep 12 2004

%E Edited, corrected and extended by _Robert G. Wilson v_ and _Klaus Brockhaus_, Sep 14 2004

%E Edited by _N. J. A. Sloane_, Jul 02 2008 at the suggestion of _R. J. Mathar_