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A084289 Primes p such that the arithmetic mean of p and the next prime after p is a true prime power from A025475.

%I A084289 #14 Oct 15 2023 11:46:39
%S A084289 3,7,61,79,619,1669,4093,822631,1324783,2411797,2588869,2778877,
%T A084289 3243589,3636631,3736477,5527189,6115717,6405943,8720191,9005989,
%U A084289 12752029,16056031,16589317,18087991,21743551,25230511,29343871,34586131,37736431,39150037,40056229
%N A084289 Primes p such that the arithmetic mean of p and the next prime after p is a true prime power from A025475.
%H A084289 Donovan Johnson, <a href="/A084289/b084289.txt">Table of n, a(n) for n = 1..1000</a>
%F A084289 Primes p(j) such that (p(j)+p(j+1))/2 = q(m)^w, where q(m) is a prime.
%e A084289 n = prime(9750374) = 174689077, next prime = 174689101, mean = 174689089 = 13217^2, a prime power. The arithmetic mean of two consecutive primes is never prime, while between two consecutive primes, prime powers occur. These prime powers are in the middle of gap: p+d/2 = q^w. The prime power is most often square and very rarely occurs more than once (see A053706).
%t A084289 fi[x_] := FactorInteger[x] ff[x_] := Length[FactorInteger[x]] Do[s=(Prime[n]+Prime[n+1])/2; s1=ff[s]; If[Equal[s1,1],Print[{n,p=Prime[n],s,fi[s],s-p,s1}]], {n,1,10000000}]
%t A084289 Select[Partition[Prime[Range[25*10^5]],2,1],PrimePowerQ[Mean[#]]&][[;;,1]] (* _Harvey P. Dale_, Oct 15 2023 *)
%Y A084289 Cf. A053706, A000961, A025475.
%K A084289 nonn
%O A084289 1,1
%A A084289 _Labos Elemer_, May 26 2003