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A138694
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Numbers n such that the set {2*n+p^2, p any prime} contains exactly one prime.
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7
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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.
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FORMULA
| {This sequence here} Union {A138685} = {A016777}.
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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.
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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]]
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CROSSREFS
| Cf. A138479, A138685, A138686, A138691, A138692, A138693, A138694, A007528, A138696, A138697, A138698, A138699, A138700.
Sequence in context: A137379 A096676 A092863 * A177965 A179209 A168565
Adjacent sequences: A138691 A138692 A138693 * A138695 A138696 A138697
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KEYWORD
| nonn
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AUTHOR
| Artur Jasinski (grafix(AT)csl.pl), Mar 27 2008
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EXTENSIONS
| Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 15 2009
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