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A192618 Prime powers p^k with even exponents k > 0 such that (1 + p^k)/2 is prime.

%I A192618 #17 Apr 17 2023 06:18:43
%S A192618 9,25,81,121,361,625,841,2401,3481,3721,5041,6241,10201,14641,17161,
%T A192618 19321,28561,32761,39601,73441,83521,121801,143641,167281,201601,
%U A192618 212521,271441,279841,323761,326041,398161,410881,436921,546121,564001,674041
%N A192618 Prime powers p^k with even exponents k > 0 such that (1 + p^k)/2 is prime.
%C A192618 Subsequence of A056798.
%C A192618 From _R. J. Mathar_, Jul 11 2011: (Start)
%C A192618 For odd k we first have the case k=1, where (1+p)/2 is either classified as A005383 or A176897.
%C A192618 For odd k >= 3, (1+p^k)/2 is not prime. [Sketch of proof: for p=2 it is not integer. Otherwise for odd k, (1+p^k)/(1+p) = Sum_{j=0..k-1} (-p)^j, an integer, so 1+p^k is a multiple of 1+p. For odd p, (1+p^k)/2 is a multiple of (1+p)/2 and therefore composite.] (End)
%H A192618 Klaus Brockhaus, <a href="/A192618/b192618.txt">Table of n, a(n) for n = 1..1000</a>
%t A192618 Select[Union[Flatten[Table[Prime[n]^k, {n, 142}, {k, 0, 32, 2}]]], PrimeQ[(# + 1)/2] &] (* _Alonso del Arte_, Jul 05 2011 *)
%o A192618 (Magma) e:=20; u:=1000; z:=Min(2^e, u^2); S:=[ q: p in PrimesUpTo(u), k in [2..e by 2] | q le z and IsEven(1+q) and IsPrime((1+q) div 2) where q is p^k ]; Sort(~S); S;
%Y A192618 Cf. A056798.
%K A192618 nonn
%O A192618 1,1
%A A192618 _Klaus Brockhaus_, Jul 05 2011