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

47961, 123201, 131769, 826281, 870489, 2486929, 3294225, 5239521, 5294601, 5774409, 6215049, 6335289, 6848689, 9308601, 10634121, 16072081, 17164449, 17732521, 18896409, 19298449, 22667121, 24413481, 25391521, 25836889, 30769209, 32569849, 33535681

Offset: 1

Author

K. D. Bajpai, Dec 22 2013

Keywords

• nonn

Examples

			47961 is in the sequence because 47961 = 219^2 = (47951+47963+47969)/3, the arithmetic mean of three consecutive primes.
131769 is in the sequence because 131769 = 363^2 = (131759+131771+131777)/3, the arithmetic mean of three consecutive primes.
		

Links

• K. D. Bajpai, Table of n, a(n) for n = 1..3650

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).

Programs

Maple

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);

Mathematica

amQ[{a_,b_,c_}]:=Module[{m=Mean[{a,b,c}]},IntegerQ[Sqrt[m]]&&aHarvey P. Dale, Mar 14 2014 *)

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

Extensions

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