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

1,1

Links

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

Example

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.

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]]&&a<m<b<c]; Mean/@Select[Partition[Prime[Range[2100000]], 3, 1], amQ] (* Harvey P. Dale, Mar 14 2014 *)

Prog

(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

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).

Sequence in context: A203929 A234172 A236716 * A043615 A252949 A203547

Adjacent sequences: A234294 A234295 A234296 * A234298 A234299 A234300

Keyword

nonn

Author

K. D. Bajpai, Dec 22 2013

Extensions

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

Status

approved