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

%I A069495 #22 Oct 23 2021 03:23:05
%S A069495 4,9,64,81,144,225,324,441,625,1089,1681,2601,3600,4096,5184,6084,
%T A069495 8464,12544,13689,16641,19044,19600,25281,27225,28224,29584,36864,
%U A069495 38025,39204,45369,46656,47524,51984,56169,74529,87025,88804,91809,92416,95481,103684
%N A069495 Squares which are the arithmetic mean of two consecutive primes.
%H A069495 Amiram Eldar, <a href="/A069495/b069495.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)
%F A069495 a(n) = (A075190(n))^2. - _Zak Seidov_
%e A069495 144 = (139 + 149)/2 is a member.
%p A069495 a:= proc(n) option remember; local k, kk, p, q;
%p A069495       for k from 1 +`if`(n=1, 1, iroot(a(n-1), 2))
%p A069495       do kk:= k^2; p, q:= prevprime(kk), nextprime(kk);
%p A069495          if (p+q)/2=kk then return kk fi
%p A069495       od
%p A069495     end:
%p A069495 seq(a(n), n=1..60);  # _Alois P. Heinz_, Dec 21 2013
%t A069495 p = -1; Do[q = Prime[n]; If[ IntegerQ[ Sqrt[(p + q)/2]], Print[(p + q)/2]]; p = q, {n, 1, 10000} ]
%t A069495 Select[Mean/@Partition[Prime[Range[11000]],2,1],IntegerQ[Sqrt[#]]&] (* _Harvey P. Dale_, Jan 23 2019 *)
%Y A069495 Cf. A062703, A075190.
%Y A069495 Intersection of A000290 and A024675.
%K A069495 nonn
%O A069495 1,1
%A A069495 _Amarnath Murthy_, Mar 30 2002
%E A069495 Edited and extended by _Robert G. Wilson v_, Apr 01 2002