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

1, 4, 7, 10, 16, 19, 22, 25, 31, 37, 40, 46, 49, 52, 61, 64, 70, 79, 82, 85, 91, 94, 109, 112, 115, 121, 124, 127, 130, 136, 142, 151, 154, 169, 172, 175, 187, 190, 196, 205, 211, 217, 220, 226, 229, 235, 241, 247, 250, 256, 274, 277, 280, 289, 292, 295, 304, 316

Offset

1,2

Comments

The sequence forms a subset of A016777, as explained below:

For each prime p<>3 we have p^2 =1 (mod 3), see A024700.

(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.

The subcase where 2*k+3^2 is prime generates this sequence here; the subcase where it is not generates A138685.

(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)

to generate a prime 2*k+p^2 = 1 (mod 3), so these k generate many primes (of the form A002476).

(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)

to generate a prime 2*k+p^2 = 2 (mod 3), so these k generate many primes (of the form A003627).

The unique primes associated with each n are in A007528: n=1 associated with A007528(2)=11=2*1+3^2,

n=4 associated with A007528(3)=17=2*4+3^2 etc.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Formula

{This sequence here} Union {A138685} = {A016777}.

Example

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,..}
contains the primes 31, 127, 367,..., generated with p=5,11,19...
4 is in the sequence because {8+2^2, 8+3^3, 8+5^2, 8+7^2,..} = {12, 17, 33, 57,...} contains
only one prime (that is, 17), generated with p=3.

Mathematica

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]]

Crossrefs

Cf. A138479, A138685, A138686, A138691, A138692, A138693, A138694, A007528, A138696, A138697, A138698, A138699, A138700.

Sequence in context: A310708 A310709 A310710 * A359274 A177965 A179209

Adjacent sequences: A138691 A138692 A138693 * A138695 A138696 A138697

Keyword

nonn

Author

Artur Jasinski, Mar 27 2008

Extensions

Edited by R. J. Mathar, May 15 2009

Status

approved