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Search: id:A137270
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| A137270 |
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Primes p such that p^2 - 6 is also prime. |
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2
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| 3, 5, 7, 13, 17, 23, 47, 53, 67, 73, 83, 97, 107, 113, 167, 193, 197, 263, 293, 317, 367, 373, 383, 457, 463, 467, 487, 503, 557, 593, 607, 643, 647, 673, 677, 683, 773, 787, 797, 823, 827, 857, 877, 887, 947, 1033, 1063, 1087, 1103, 1187, 1193, 1223, 1303
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Each of the primes p = 2,3,5,7,13 has the property that the quadratic polynomial phi(x) = x^2 + x - p^2 takes on only prime values for x = 1,2,...,2p-2; each case giving exactly one repetition, in phi(p-1) = -p and phi(p) = p.
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REFERENCES
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F. G. Frobenius, Uber quadratische Formen, die viele Primzahlen darstellen, Sitzungsber. d. Konigl. Acad. d. Wiss. zu Berlin, 1912, 966 - 980.
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FORMULA
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A000040 INTERSECT A028879. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 16 2008
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EXAMPLE
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The (2 x 7 - 2) -1 = 11 primes given by the polynomial x^2 + x - 7^2 for x = 1, 2, ..., 2 x 7 - 2 are -47, -43, -37, -29, -19, -7, 7, 23, 41, 61, 83, 107.
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MAPLE
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isA028879 := proc(n) isprime(n^2-6) ; end: isA137270 := proc(n) isprime(n) and isA028879(n) ; end: for i from 1 to 300 do if isA137270(ithprime(i)) then printf("%d, ", ithprime(i)) ; fi ; od: - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 16 2008
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MATHEMATICA
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f[n_]:=n^2-6; lst={}; Do[p=Prime[n]; If[PrimeQ[f[p]], AppendTo[lst, p]], {n, 7!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jul 16 2009]
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CROSSREFS
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Cf. A062326.
Sequence in context: A067567 A163998 A104294 this_sequence A071111 A038929 A070806
Adjacent sequences: A137267 A137268 A137269 this_sequence A137271 A137272 A137273
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KEYWORD
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nonn
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AUTHOR
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Ben de la Rosa and Johan Meyer (meyerjh.sci(AT)ufa.ac.za), Mar 13 2008
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EXTENSIONS
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Corrected and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 16 2008
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