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%I #22 Jun 25 2022 17:08:23
%S 15876,35721,59049,65025,488601,828100,1144900,3857296,4822416,
%T 4901796,5107600,5322249,5856400,6100900,6760000,10536516,11716929,
%U 12503296,13468900,14197824,14638276,15163236,18748900,21455424,22127616,22638564,24049216,24098281,24108100
%N Squares t^2 = (p+q+r+s)/4 which are the arithmetic mean of four consecutive primes such that p < t^2 < q < r < s.
%H K. D. Bajpai, <a href="/A234318/b234318.txt">Table of n, a(n) for n = 1..2869</a>
%e 15876 is in the sequence because 15876 = 126^2 = (15859+15877+15881+15887)/4, the arithmetic mean of four consecutive primes.
%e 35721 is in the sequence because 35721 = 189^2 = (35677+35729+35731+35747)/4, the arithmetic mean of four consecutive primes.
%p KD := proc() local a,b,d,e,f,g; a:=n^2;b:=prevprime(a); d:=nextprime(a); e:=nextprime(d); f:=nextprime(e); g:=(b+d+e+f)/4; if a=g then RETURN (a); fi; end: seq(KD(), n=2..10000);
%t fcpQ[{a_,b_,c_,d_}]:=Module[{m=Mean[{a,b,c,d}]},IntegerQ[ Sqrt[ m]] && a< m< b]; Mean/@Select[Partition[Prime[Range[1600000]],4,1],fcpQ] (* _Harvey P. Dale_, Apr 24 2017 *)
%o (PARI) list(lim)=my(v=List(), p=2, q=3, r=5, t); forprime(s=7, nextprime(nextprime(lim+1)+1), t=(p+q+r+s)/4; if(denominator(t)==1 && issquare(t) && t < q, listput(v, t)); p=q; q=r; r=s); Vec(v) \\ _Charles R Greathouse IV_, Jan 03 2014
%Y Cf. A000290 (squares: a(n) = n^2).
%Y Cf. A062703 (squares: sum of two consecutive primes).
%Y Cf. A069495 (squares: arithmetic mean of two consecutive primes).
%Y Cf. A234240 (cubes: arithmetic mean of three consecutive primes).
%Y Cf. A234297 (squares: arithmetic mean of three consecutive primes).
%K nonn
%O 1,1
%A _K. D. Bajpai_, Dec 23 2013
%E Definition corrected by _Michel Marcus_ and _Charles R Greathouse IV_, Jan 03 2014
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