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A069495 Squares which are the arithmetic mean of two consecutive primes. 12

%I

%S 4,9,64,81,144,225,324,441,625,1089,1681,2601,3600,4096,5184,6084,

%T 8464,12544,13689,16641,19044,19600,25281,27225,28224,29584,36864,

%U 38025,39204,45369,46656,47524,51984,56169,74529,87025,88804,91809,92416,95481,103684

%N Squares which are the arithmetic mean of two consecutive primes.

%H Alois P. Heinz, <a href="/A069495/b069495.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = (A075190(n))^2. - _Zak Seidov_

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

%p a:= proc(n) option remember; local k, kk, p, q;

%p for k from 1 +`if`(n=1, 1, iroot(a(n-1), 2))

%p do kk:= k^2; p, q:= prevprime(kk), nextprime(kk);

%p if (p+q)/2=kk then return kk fi

%p od

%p end:

%p seq(a(n), n=1..60); # _Alois P. Heinz_, Dec 21 2013

%t p = -1; Do[q = Prime[n]; If[ IntegerQ[ Sqrt[(p + q)/2]], Print[(p + q)/2]]; p = q, {n, 1, 10000} ]

%t Select[Mean/@Partition[Prime[Range[11000]],2,1],IntegerQ[Sqrt[#]]&] (* _Harvey P. Dale_, Jan 23 2019 *)

%Y Cf. A062703.

%K nonn

%O 1,1

%A _Amarnath Murthy_, Mar 30 2002

%E Edited and extended by _Robert G. Wilson v_, Apr 01 2002

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Last modified November 11 22:31 EST 2019. Contains 329046 sequences. (Running on oeis4.)