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

1,1

Links

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

Formula

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

Example

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

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 *)

Crossrefs

Cf. A062703, A075190.

Intersection of A000290 and A024675.

Sequence in context: A369866 A028822 A270696 * A029788 A062826 A201918

Adjacent sequences: A069492 A069493 A069494 * A069496 A069497 A069498

Keyword

nonn

Author

Amarnath Murthy, Mar 30 2002

Extensions

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

Status

approved