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Numbers n such that the mean of 5 consecutive squares starting with n^2 is prime.
4

%I #23 Feb 04 2024 03:19:33

%S 1,7,13,19,31,37,43,55,79,97,103,109,115,121,145,169,217,223,235,241,

%T 247,253,271,295,301,307,319,343,349,361,367,373,385,415,421,427,439,

%U 445,451,475,499,511,547,553,559,571,601,607,649,673,679,697,709,751

%N Numbers n such that the mean of 5 consecutive squares starting with n^2 is prime.

%C Sum of 5 consecutive squares starting with n^2 is equal to 5*(6 + 4*n + n^2) and mean is (6 + 4*n + n^2) = (n+2)^2 + 2. Hence a(n) = A067201(n+2).

%C Also, numbers n such that A000217(n) + A000217(n+3) is prime. - _Bruno Berselli_, Apr 17 2013

%H Bruno Berselli, <a href="/A129389/b129389.txt">Table of n, a(n) for n = 1..1000</a>

%e (1^2 + ... + 5^2)/5 = 11, which is prime;

%e (7^2 + ... + 11^2)/5 = 83, which is prime;

%e (13^2 + ... + 17^2)/5 = 227, which is prime.

%t Select[Range[800], PrimeQ[#^2 + 4 # + 6] &] (* _Bruno Berselli_, Apr 17 2012 *)

%o (Magma) [n: n in [1..800] | IsPrime(n^2+4*n+6)]; /* or, from the second comment: */ A000217:=func<i | i*(i+1) div 2>; [n: n in [1..800] | IsPrime(A000217(n)+A000217(n+3))]; // _Bruno Berselli_, Apr 17 2013

%o (SageMath) [n for n in (1..1000) if is_prime(n^2+4*n+6)] # _G. C. Greubel_, Feb 04 2024

%Y Cf. A056899, A067201, A129388.

%Y Cf. A000217, A128815 (numbers n such that A000217(n)+A000217(n+2) is prime). [_Bruno Berselli_, Apr 17 2013]

%K nonn

%O 1,2

%A _Zak Seidov_, Apr 12 2007