A069495 Squares which are the arithmetic mean of two consecutive primes.

4, 9, 64, 81, 144, 225, 324, 441, 625, 1089, 1681, 2601, 3600, 4096, 5184, 6084, 8464, 12544, 13689, 16641, 19044, 19600, 25281, 27225, 28224, 29584, 36864, 38025, 39204, 45369, 46656, 47524, 51984, 56169, 74529, 87025, 88804, 91809, 92416, 95481, 103684

Offset: 1

Author

Amarnath Murthy, Mar 30 2002

Keywords

• nonn

Examples

			144 = (139 + 149)/2 is a member.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Crossrefs

Cf. A062703, A075190.

Intersection of A000290 and A024675.

Programs

Maple

a:= proc(n) option remember; local k, kk, p, q;
      for k from 1 +`if`(n=1, 1, iroot(a(n-1), 2))
      do kk:= k^2; p, q:= prevprime(kk), nextprime(kk);
         if (p+q)/2=kk then return kk fi
      od
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Dec 21 2013

Mathematica

p = -1; Do[q = Prime[n]; If[ IntegerQ[ Sqrt[(p + q)/2]], Print[(p + q)/2]]; p = q, {n, 1, 10000} ]
Select[Mean/@Partition[Prime[Range[11000]],2,1],IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Jan 23 2019 *)

Formula

a(n) = (A075190(n))^2. - Zak Seidov

Extensions

Edited and extended by Robert G. Wilson v, Apr 01 2002