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%I #28 Aug 13 2015 08:45:47
%S 8,9,25,32,121,289,841,1681,2048,3481,5041,10201,11449,18769,22201,
%T 32041,36481,38809,51529,57121,72361,78961,96721,120409,131072,175561,
%U 185761,212521,271441,323761,358801,380689,410881,434281,654481,674041,683929,734449
%N Perfect powers m^k such that m, k and m+k are primes.
%C Necessarily either m or k = 2, thus if a(n) is even, it is a power of 2 with odd prime exponent, otherwise (if a(n) is odd), it is a square of odd prime.
%C For each term m^k, there will be another k^m.
%C a(3), a(5), a(11) are of the form n! + 1.
%C Let F(m,k) = m*k, such that m^k = a(n), so A108605 is a subsequence of F. For example a(1) = 2^3 and F(2,3) = A108605(1).
%e a(1) = 8, because 8 = 2^3 and 2+3 = 5.
%e a(4) = 32, because 32 = 2^5 and 2+5 = 7.
%e a(5) = 121, because 121 = 11^2 and 11+2 = 13.
%e a(25) = 131072, because 131072 = 2^17 and 2+17 = 19.
%t SmallestDivisor[n_] := If[n == 1, 1, Divisors[n][[2]]]; perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; ppl = Select[Range[200000], perfectPowerQ]; base[n_] := ppl[[n]]^(1/exp[n]); exp[n_] := SmallestDivisor[GCD @@ FactorInteger[ppl[[n]]][[All, 2]] ]; pp2l = Table[ {base[n], exp[n]}, {n, Length[ppl]}]; p[n_] := pp2l[[n]][[1]]; q[n_] := pp2l[[n]][[2]]; lt = Select[Range[Length[pp2l]], PrimeQ[p[#]] && PrimeQ[q[#]] && PrimeQ[p[#] + q[#]] &]; ppl[[lt]]
%t Select[Range[10^6], Length[f = FactorInteger@ #] == 1 && PrimeQ@ f[[1, 2]] && PrimeQ@ Total@ f[[1]] &] (* _Giovanni Resta_, Jun 23 2015 *)
%Y Subsequence of A001597, A000961.
%Y Cf. A000079, A001248, A108605, A109611.
%K nonn
%O 1,1
%A _Carlos Eduardo Olivieri_, May 28 2015
%E a(28)-a(38) from _Giovanni Resta_, Jun 23 2015
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