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%I #14 Jun 22 2022 10:54:13
%S 4,9,64,81,144,225,324,441,625,1089,1681,1728,2601,3600,4096,5184,
%T 6084,8464,12544,13689,16641,17576,19044,19600,21952,25281,27225,
%U 28224,29584,36864,38025,39204,45369,46656,47524,51984,56169,74529,87025,88804,91809,92416,95481
%N Perfect powers that are the average of two consecutive primes.
%C The corresponding lesser primes are in A242380. - _Michel Marcus_, Mar 23 2016
%H Charles R Greathouse IV, <a href="/A270696/b270696.txt">Table of n, a(n) for n = 1..10000</a>
%e 1728 is a term because 1728 = 12^3 = (1723 + 1733)/2.
%t Select[Mean/@Partition[Prime[Range[10000]],2,1],GCD@@FactorInteger[#][[All,2]]>1&] (* _Harvey P. Dale_, Jun 22 2022 *)
%o (PARI) for(n=2, 1e5, if((nextprime(n) - n) == (n - precprime(n)) && ispower(n), print1(n, ", ")));
%o (PARI) list(lim)=my(v=List(),t); forprime(e=2,logint(lim\=1,2), for(m=2,sqrtnint(lim,e), t=m^e; if(t-precprime(t)==nextprime(t)-t, listput(v,t)))); Set(v) \\ _Charles R Greathouse IV_, Mar 21 2016
%Y Cf. A001597, A024675, A242380.
%K nonn
%O 1,1
%A _Altug Alkan_, Mar 21 2016
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