%I #16 Jan 12 2019 19:33:01

%S 3,7,61,79,139,223,317,439,619,1087,1669,2593,3593,4093,5179,6079,

%T 8461,12541,13687,16633,19037,19597,25261,27211,28219,29581,36857,

%U 38011,39199,45361,46649,47521,51977,56167,74527,87013,88801,91807,92413,95479,103681

%N Primes p such that (p+nextprime(p))/2 is a perfect square.

%C Primes for which average of two consecutive primes is a perfect square.

%H Charles R Greathouse IV, <a href="/A225195/b225195.txt">Table of n, a(n) for n = 1..5000</a>

%e 61 is in the list since (61+67)/2=64=8^2.

%t Select[Prime[Range[10000]],IntegerQ[Sqrt[(# + NextPrime[#])/2]] &]

%t Select[Partition[Prime[Range[10000]],2,1],IntegerQ[Sqrt[Mean[#]]]&][[All,1]] (* _Harvey P. Dale_, Jan 12 2019 *)

%o (PARI) p=2;forprime(q=3,1e9,if(issquare((p+q)/2),print1(p", "));p=q) \\ _Charles R Greathouse IV_, May 09 2013

%Y Cf. A000040, A000290, A069495, A075190

%K nonn

%O 1,1

%A _Jayanta Basu_, May 09 2013