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A138694 Numbers n such that the set {2*n+p^2, p any prime} contains exactly one prime.

%I A138694 #8 May 21 2013 02:29:36
%S A138694 1,4,7,10,16,19,22,25,31,37,40,46,49,52,61,64,70,79,82,85,91,94,109,
%T A138694 112,115,121,124,127,130,136,142,151,154,169,172,175,187,190,196,205,
%U A138694 211,217,220,226,229,235,241,247,250,256,274,277,280,289,292,295,304,316
%N A138694 Numbers n such that the set {2*n+p^2, p any prime} contains exactly one prime.
%C A138694 The sequence forms a subset of A016777, as explained below:
%C A138694 For each prime p<>3 we have p^2 =1 (mod 3), see A024700.
%C A138694 (i) For the k where 2*k=2 (mod 3), that is where k=1 (mod 3), this leads to 2*k+p^2=0 (mod 3), so the 2*k+p^2 are divisible by 3 (not prime) unless p=3.
%C A138694 The subcase where 2*k+3^2 is prime generates this sequence here; the subcase where it is not generates A138685.
%C A138694 (ii) For the k where 2*k=0 (mod 3), that is where k=0 (mod 3), one can select any p^2 =1 (mod 3)
%C A138694 to generate a prime 2*k+p^2 = 1 (mod 3), so these k generate many primes (of the form A002476).
%C A138694 (iii) For the k where 2*k=1 (mod 3), that is where k=2 (mod 3), one can select any p^2 =1 (mod 3)
%C A138694 to generate a prime 2*k+p^2 = 2 (mod 3), so these k generate many primes (of the form A003627).
%C A138694 The unique primes associated with each n are in A007528: n=1 associated with A007528(2)=11=2*1+3^2,
%C A138694 n=4 associated with A007528(3)=17=2*4+3^2 etc.
%H A138694 Vincenzo Librandi, <a href="/A138694/b138694.txt">Table of n, a(n) for n = 1..1000</a>
%F A138694 {This sequence here} Union {A138685} = {A016777}.
%e A138694 3 is not in the sequence because {6+2^2, 6+3^3, 6+5^2, 6+7^2,..} = {10, 15, 31, 55,..,127,..,367,..}
%e A138694 contains the primes 31, 127, 367,..., generated with p=5,11,19...
%e A138694 4 is in the sequence because {8+2^2, 8+3^3, 8+5^2, 8+7^2,..} = {12, 17, 33, 57,...} contains
%e A138694 only one prime (that is, 17), generated with p=3.
%t A138694 b = {}; Do[a = {}; Do[If[PrimeQ[2*k + Prime[n]^2], AppendTo[a, k]], {n, 1, 100}]; If[Length[a] < 2, AppendTo[b, a]], {k, 1, 500}]; Union[Flatten[b]]
%Y A138694 Cf. A138479, A138685, A138686, A138691, A138692, A138693, A138694, A007528, A138696, A138697, A138698, A138699, A138700.
%K A138694 nonn
%O A138694 1,2
%A A138694 _Artur Jasinski_, Mar 27 2008
%E A138694 Edited by _R. J. Mathar_, May 15 2009