A049423 Primes of the form k^2 + 3.

3, 7, 19, 67, 103, 199, 487, 787, 1447, 2503, 2707, 3847, 4099, 4903, 5479, 5779, 8467, 8839, 11239, 12547, 14887, 16903, 17959, 19603, 21319, 23719, 24967, 25603, 29587, 31687, 47527, 52903, 58567, 59539, 61507, 65539, 75079, 81799, 88807

Offset: 1

Author

Paul Jobling (paul.jobling(AT)whitecross.com)

Keywords

• easy
• nonn

Comments

Note that all terms after the first are congruent to 7 modulo 12.

Examples

			19 is prime and is equal to 4^2 + 3, so 19 is a term.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Near-Square Prime

Crossrefs

Cf. A002496, A056899. Note that apart from first term, all of (a(n)-7)/12 have to be terms of A001082 for a(n) to be prime.

Programs

Magma

[n: n in PrimesUpTo(175000) | IsSquare(n-3)];  // Bruno Berselli, Apr 05 2011

Magma

[a: n in [0..300] | IsPrime(a) where a is n^2+3]; // Vincenzo Librandi, Dec 08 2011

Mathematica

Intersection[Table[n^2+3,{n,0,10^2}],Prime[Range[9*10^3]]] ...or... For[i=3,i<=3,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 *)
Select[Table[n^2+3,{n,0,198000}],PrimeQ] (* Vincenzo Librandi, Dec 08 2011 *)

PARI

list(lim)=my(v=List(),t); forstep(k=0,sqrtint(lim\1-3),2, if(isprime(t=k^2+3), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Nov 06 2024

Formula

Primes m such that m-3 is a square.

For n>0, a(n) = 36*A056902(n-1)^2 + 24*A056902(n-1) + 7. - Henry Bottomley, Jul 06 2000

a(n) = 3 + (2*A097697(n-1))^2. - R. J. Mathar, Aug 07 2008

a(n) >> n^2 log n. - Charles R Greathouse IV, Nov 06 2024