A005473 Primes of form k^2 + 4.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Examples

			a(2)=29 since 29=5^2+4 is prime.
		

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

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.

Programs

Haskell

a005473 n = a005473_list !! (n-1)
a005473_list = filter ((== 1) . a010051') $ map (+ 4) a000290_list
-- Reinhard Zumkeller, Mar 12 2012

Magma

[a: n in [0..300] | IsPrime(a) where a is n^2+4]; // Vincenzo Librandi, Nov 30 2011

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 *)

PARI

for(n=1,1e3,if(isprime(t=n^2+4),print1(t","))) \\ Charles R Greathouse IV, Jul 05 2011

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

Extensions

More terms and additional comments from Henry Bottomley, Jul 06 2000