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%I M3830 #75 Sep 08 2022 08:44:33
%S 5,13,29,53,173,229,293,733,1093,1229,1373,2029,2213,3253,4229,4493,
%T 5333,7229,7573,9029,9413,10613,13229,13693,15629,18229,18773,21613,
%U 24029,26573,27893,31333,33493,37253,41213,42853,46229,47093,54293
%N Primes of form k^2 + 4.
%C a(n) mod 24 = 5 or 13 and if a(n) mod 24 =13 then a(n) mod 72 = 13.
%C From _Artur Jasinski_, Oct 30 2008: (Start)
%C Primes p such that the continued fraction of (1+sqrt(p))/2 has period 1.
%C Primes in A078370 = primes of the form 4*k^2 + 4*k + 5 = (2*k+1)^2 + 4.
%C (End)
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A005473/b005473.txt">Table of n, a(n) for n = 1..4600</a>
%H D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1974-0352049-8">The simplest cubic fields</a>, Math. Comp., 28 (1974), 1137-1152.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Near-SquarePrime.html">Near-Square Prime</a>
%F 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).
%F For n>=2, a(n) = A098062(n-1). - _Zak Seidov_, Apr 12 2007
%e a(2)=29 since 29=5^2+4 is prime.
%p select(isprime,[seq(4*k^2 + 4*k + 5, k=0..1000)]); # _Robert Israel_, Nov 02 2014
%t 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 *)
%t 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 *)
%t Select[Table[n^2+4,{n,0,7000}],PrimeQ] (* _Vincenzo Librandi_, Nov 30 2011 *)
%o (PARI) for(n=1,1e3,if(isprime(t=n^2+4),print1(t","))) \\ _Charles R Greathouse IV_, Jul 05 2011
%o (Magma) [a: n in [0..300] | IsPrime(a) where a is n^2+4]; // _Vincenzo Librandi_, Nov 30 2011
%o (Haskell)
%o a005473 n = a005473_list !! (n-1)
%o a005473_list = filter ((== 1) . a010051') $ map (+ 4) a000290_list
%o -- _Reinhard Zumkeller_, Mar 12 2012
%Y Subsequence of A185086.
%Y 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.
%Y Cf. A146326, A010051, A000290, A138353, A098062.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_
%E More terms and additional comments from _Henry Bottomley_, Jul 06 2000
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