A005473 Primes of form k^2 + 4. (Formerly M3830)

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

1,1

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: A247903 A350687 A240130 * A086732 A162329 A299895

Adjacent sequences: A005470 A005471 A005472 * A005474 A005475 A005476

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

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

Status

approved