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A005473
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Primes of form k^2 + 4.
(Formerly M3830)
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18
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5, 13, 29, 53, 173, 229, 293, 733, 1093, 1229, 1373, 2029, 2213, 3253, 4229, 4493, 5333, 7229, 7573, 9029, 9413, 10613, 13229, 13693, 15629, 18229, 18773, 21613, 24029, 26573, 27893, 31333, 33493, 37253, 41213, 42853, 46229, 47093, 54293
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OFFSET
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1,1
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COMMENTS
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a(n) mod 24 = 5 or 13 and if a(n) mod 24 =13 then a(n) mod 72 = 13.
Primes p such that the continued fraction of (1+sqrt(p))/2 has period 1.
Primes in A078370 = primes of the form 4*k^2 + 4*k + 5 = (2*k+1)^2 + 4.
(End)
Starting at a(3) all the primes in this sequence can be expressed as the following sum: ((2*k+1)*(2*k+3)+(2*k+3)*(2*k+5)+(2*k+5)+(2*k+7)+(2*k+7)*(2*k+9))/4 for some values (not all!) of k>=0. Thus for a(5)=173 the sum is (9*11 + 11*13 + 13*15 + 15*17)/4=173. - J. M. Bergot, Nov 03 2014
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = 24*A056904(n)+m, where m=13 if A056904(n) is three times a triangular number (and n>0) and m=5 if A056904(n) is not three times a triangular number (or n=0).
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EXAMPLE
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a(2)=29 since 29=5^2+4 is prime.
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MAPLE
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select(isprime, [seq(4*k^2 + 4*k + 5, k=0..1000)]); # Robert Israel, Nov 02 2014
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MATHEMATICA
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Intersection[Table[n^2+4, {n, 0, 10^2}], Prime[Range[9*10^3]]] ...or... For[i=4, i<=4, a={}; Do[If[PrimeQ[n^2+i], AppendTo[a, n^2+i]], {n, 0, 100}]; Print["n^2+", i, ", ", a]; i++ ] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
aa = {}; Do[If[PrimeQ[4 k^2 + 4 k + 5], AppendTo[aa, 4 k^2 + 4 k + 5]], {k, 0, 200}]; aa (* Artur Jasinski, Oct 30 2008 *)
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PROG
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(Magma) [a: n in [0..300] | IsPrime(a) where a is n^2+4]; // Vincenzo Librandi, Nov 30 2011
(Haskell)
a005473 n = a005473_list !! (n-1)
a005473_list = filter ((== 1) . a010051') $ map (+ 4) a000290_list
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CROSSREFS
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a(n)-4 is contained in A016754. (a(n)-5)/8 is contained in A000217. Either (a(n)-5)/24 is contained in A001318 (if a(n) mod 24=5) or (a(n)-13)/72 is contained in A000217 (if a(n) mod 24=13). Floor[a(n)/24] is contained in A001840.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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