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A234297 Squares t^2 = (p+q+r)/3 which are the arithmetic mean of three consecutive primes such that p < t^2 < q < r.

%I A234297 #27 Jun 25 2022 17:08:27
%S A234297 47961,123201,131769,826281,870489,2486929,3294225,5239521,5294601,
%T A234297 5774409,6215049,6335289,6848689,9308601,10634121,16072081,17164449,
%U A234297 17732521,18896409,19298449,22667121,24413481,25391521,25836889,30769209,32569849,33535681
%N A234297 Squares t^2 = (p+q+r)/3 which are the arithmetic mean of three consecutive primes such that p < t^2 < q < r.
%H A234297 K. D. Bajpai, <a href="/A234297/b234297.txt">Table of n, a(n) for n = 1..3650</a>
%e A234297 47961 is in the sequence because 47961 = 219^2 = (47951+47963+47969)/3, the arithmetic mean of three consecutive primes.
%e A234297 131769 is in the sequence because 131769 = 363^2 = (131759+131771+131777)/3, the arithmetic mean of three consecutive primes.
%p A234297 with(numtheory):KD := proc() local a,b,d,e,f; a:=n^2; b:=prevprime(a); d:=nextprime(a); e:=nextprime(d); f:=(b+d+e)/3; if a=f then RETURN (a); fi; end: seq(KD(), n=2..10000);
%t A234297 amQ[{a_,b_,c_}]:=Module[{m=Mean[{a,b,c}]},IntegerQ[Sqrt[m]]&&a<m<b<c]; Mean/@Select[Partition[Prime[Range[2100000]],3,1],amQ] (* _Harvey P. Dale_, Mar 14 2014 *)
%o A234297 (PARI) list(lim)=my(v=List(),p=2,q=3,t); forprime(r=5, nextprime(nextprime(lim+1)+1), t=(p+q+r)/3; if(denominator(t)==1 && issquare(t) && t < q, listput(v, t)); p=q;q=r); Vec(v) \\ _Charles R Greathouse IV_, Jan 03 2014
%Y A234297 Cf. A000290 (squares:  a(n) = n^2).
%Y A234297 Cf. A062703 (squares: sum of two consecutive primes).
%Y A234297 Cf. A069495 (squares: arithmetic mean of two consecutive primes).
%Y A234297 Cf. A234240 (cubes: arithmetic mean of three consecutive primes).
%K A234297 nonn
%O A234297 1,1
%A A234297 _K. D. Bajpai_, Dec 22 2013
%E A234297 Definition corrected by _Michel Marcus_ and _Charles R Greathouse IV_, Jan 03 2014