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 A005473 Primes of form n^2 + 4. (Formerly M3830) 15
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS a(n) mod 24 = 5 or 13 and if a(n) mod 24 =13 then a(n) mod 72 = 13. From Artur Jasinski, Oct 30 2008: (Start) 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 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..4600 D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152. Eric Weisstein's World of Mathematics, Near-Square Prime FORMULA 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). For n>=2, a(n) = A098062(n-1). - Zak Seidov, Apr 12 2007 EXAMPLE a(2)=29 since 29=5^2+4 is prime. MAPLE select(isprime, [seq(4*k^2 + 4*k + 5, k=0..1000)]); # Robert Israel, Nov 02 2014 MATHEMATICA 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 *) Select[Table[n^2+4, {n, 0, 7000}], PrimeQ] (* Vincenzo Librandi, Nov 30 2011 *) PROG (PARI) for(n=1, 1e3, if(isprime(t=n^2+4), print1(t", "))) \\ Charles R Greathouse IV, Jul 05 2011 (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 -- Reinhard Zumkeller, Mar 12 2012 CROSSREFS Subsequence of A185086. 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. Cf. A146326, A010051, A000290, A138353, A098062. Sequence in context: A078370 A247903 A240130 * A086732 A162329 A299895 Adjacent sequences:  A005470 A005471 A005472 * A005474 A005475 A005476 KEYWORD nonn,easy AUTHOR EXTENSIONS More terms and additional comments from Henry Bottomley, Jul 06 2000 STATUS approved

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Last modified May 23 12:38 EDT 2019. Contains 323514 sequences. (Running on oeis4.)