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A056899
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Primes of the form n^2+2.
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30
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2, 3, 11, 83, 227, 443, 1091, 1523, 2027, 3251, 6563, 9803, 11027, 12323, 13691, 15131, 21611, 29243, 47963, 50627, 56171, 59051, 62003, 65027, 74531, 88211, 91811, 95483, 103043, 119027, 123203, 131771, 136163, 140627, 149771, 173891
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Note that all terms after the first two are equal to 11 modulo 72 and that (a(n)-11)/72 is a triangular number, since they have to be 2 more than the square of an odd multiple of 3 to be prime and if k=6m+3 then a(n)=k^2+2=72m(m+1)/2+11.
The quotient cycle length is 2 in the continued fraction expansion of sqrt(p) for these primes. E.g.: cfrac(sqrt(6563),6)= 81+1/(81+1/(162+1/(81+1/(162+1/(81+1/(162+`...`)))))) - Labos E. (labos(AT)ana.sote.hu), Feb 22 2001
Primes in A059100; except for a(2)=3 a subsequence of A007491 and congruent to 2 modulo 9. For n>2, a(n)=11 (mod 72). [From M. F. Hasler (www.univ-ag.fr/~mhasler), Apr 05 2009]
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REFERENCES
| M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein's World of Mathematics, Near-Square Prime
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FORMULA
| For n>1, a(n)=72*A000217(A056900(n-2))+11
Also, primes of form n^2 - 2n + 3.
a(n)=A067201(n)^2+2. [From M. F. Hasler (www.univ-ag.fr/~mhasler), Apr 05 2009]
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MATHEMATICA
| Intersection[Table[n^2+2, {n, 0, 10^2}], Prime[Range[9*10^3]]] ...or... For[i=2, i<=2, a={}; Do[If[PrimeQ[n^2+i], AppendTo[a, n^2+i]], {n, 0, 100}]; Print["n^2+", i, ", ", a]; i++ ] - Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 29 2008
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PROG
| (MAGMA) [n: n in PrimesUpTo(175000) | IsSquare(n-2)]; // Bruno Berselli, Apr 05 2011
(MAGMA) [ a: n in [0..450] | IsPrime(a) where a is n^2 +2 ]; // Vincenzo Librandi, Apr 06 2011
(PARI) print1("2, 3"); forstep(n=3, 1e4, 6, if(isprime(t=n^2+2), print1(", "t))) \\ Charles R Greathouse IV, Jul 19 2011
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CROSSREFS
| Cf. A002496.
Sequence in context: A008510 A042165 A089921 * A117699 A065378 A161721
Adjacent sequences: A056896 A056897 A056898 * A056900 A056901 A056902
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KEYWORD
| nonn
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Jul 05 2000
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