A234318 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.

15876, 35721, 59049, 65025, 488601, 828100, 1144900, 3857296, 4822416, 4901796, 5107600, 5322249, 5856400, 6100900, 6760000, 10536516, 11716929, 12503296, 13468900, 14197824, 14638276, 15163236, 18748900, 21455424, 22127616, 22638564, 24049216, 24098281, 24108100

Offset: 1

Author

K. D. Bajpai, Dec 23 2013

Keywords

• nonn

Examples

			15876 is in the sequence because 15876 = 126^2 = (15859+15877+15881+15887)/4, the arithmetic mean of four consecutive primes.
35721 is in the sequence because 35721 = 189^2 = (35677+35729+35731+35747)/4, the arithmetic mean of four consecutive primes.
		

Links

• K. D. Bajpai, Table of n, a(n) for n = 1..2869

Crossrefs

Cf. A000290 (squares: a(n) = n^2).

Cf. A062703 (squares: sum of two consecutive primes).

Cf. A069495 (squares: arithmetic mean of two consecutive primes).

Cf. A234240 (cubes: arithmetic mean of three consecutive primes).

Cf. A234297 (squares: arithmetic mean of three consecutive primes).

Programs

Maple

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);

Mathematica

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 *)

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

Extensions

Definition corrected by Michel Marcus and Charles R Greathouse IV, Jan 03 2014