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A267945 Primes that are a prime power plus two.

%I A267945 #19 Jan 26 2016 14:32:05
%S A267945 5,7,11,13,19,29,31,43,61,73,83,103,109,127,139,151,181,193,199,229,
%T A267945 241,271,283,313,349,421,433,463,523,571,601,619,643,661,811,823,829,
%U A267945 859,883
%N A267945 Primes that are a prime power plus two.
%C A267945 The term 'prime power' refers to the elements of A246655.
%C A267945 If we were to extend the definition of prime power to include 1, then 3 would be the first term of the sequence, because 3 = 2^0 + 2.
%C A267945 The sequence is probably infinite, since it includes all the terms of A006512 (Greater of twin primes).
%C A267945 From _Robert Israel_, Jan 22 2016: (Start)
%C A267945 Since 3 divides p or p^k+2 if k is even, the only terms of the form p^k+2 where k is even are A228034.
%C A267945 All terms not in A057735 are congruent to 1 mod 3.
%C A267945 The generalized Bunyakovsky conjecture implies that for any odd k, there are infinitely many terms of the form p^k+2. (End)
%H A267945 Robert Israel, <a href="/A267945/b267945.txt">Table of n, a(n) for n = 1..10000</a>
%H A267945 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bunyakovsky_conjecture#Generalized_Bunyakovsky_conjecture">Generalized Bunyakovsky conjecture</a>
%e A267945 5 is in the sequence because 5 = 3^1 + 2.
%e A267945 7 is in the sequence because 7 = 5^1 + 2.
%e A267945 11 is in the sequence because 11 = 3^2 + 2.
%e A267945 13 is in the sequence because 13 = 11^1 + 2.
%e A267945 29 is in the sequence because 29 = 3^3 + 2.
%p A267945 select(t -> isprime(t) and nops(numtheory:-factorset(t-2))=1, [ seq(i,i=3..1000, 2)]); # _Robert Israel_, Jan 22 2016
%t A267945 A267945Q = PrimeQ@# && (Length@# == 1 && #[[1, 1]] > 1 &@FactorInteger[# - 2]) & (* _JungHwan Min_, Jan 25 2016 *)
%t A267945 Select[Array[Prime, 100], Length@# == 1 && #[[1, 1]] > 1 &@FactorInteger[# - 2] &] (* _JungHwan Min_, Jan 25 2016 *)
%o A267945 (Sage) filter( is_prime, [ n+2 for n in prime_powers( 1, 1000 ) ] )
%o A267945 (PARI) lista(nn) = {forprime(p=2, nn, if (isprimepower(p-2), print1(p, ", ")););} \\ _Michel Marcus_, Jan 22 2016
%Y A267945 Cf. A000961, A057735, A228034, A246655, A267944.
%K A267945 nonn
%O A267945 1,1
%A A267945 _Robert C. Lyons_, Jan 22 2016