cp's OEIS Frontend

A049423 Primes of the form k^2 + 3.

%I A049423 #25 Feb 16 2025 08:32:40
%S A049423 3,7,19,67,103,199,487,787,1447,2503,2707,3847,4099,4903,5479,5779,
%T A049423 8467,8839,11239,12547,14887,16903,17959,19603,21319,23719,24967,
%U A049423 25603,29587,31687,47527,52903,58567,59539,61507,65539,75079,81799,88807
%N A049423 Primes of the form k^2 + 3.
%C A049423 Note that all terms after the first are congruent to 7 modulo 12.
%H A049423 Vincenzo Librandi, <a href="/A049423/b049423.txt">Table of n, a(n) for n = 1..10000</a>
%H A049423 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Near-SquarePrime.html">Near-Square Prime</a>
%F A049423 Primes m such that m-3 is a square.
%F A049423 For n>0, a(n) = 36*A056902(n-1)^2 + 24*A056902(n-1) + 7. - _Henry Bottomley_, Jul 06 2000
%F A049423 a(n) = 3 + (2*A097697(n-1))^2. - _R. J. Mathar_, Aug 07 2008
%F A049423 a(n) >> n^2 log n. - _Charles R Greathouse IV_, Nov 06 2024
%e A049423 19 is prime and is equal to 4^2 + 3, so 19 is a term.
%t A049423 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 *)
%t A049423 Select[Table[n^2+3,{n,0,198000}],PrimeQ] (* _Vincenzo Librandi_, Dec 08 2011 *)
%o A049423 (Magma) [n: n in PrimesUpTo(175000) | IsSquare(n-3)];  // _Bruno Berselli_, Apr 05 2011
%o A049423 (Magma) [a: n in [0..300] | IsPrime(a) where a is n^2+3]; // _Vincenzo Librandi_, Dec 08 2011
%o A049423 (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
%Y A049423 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.
%K A049423 easy,nonn
%O A049423 1,1
%A A049423 Paul Jobling (paul.jobling(AT)whitecross.com)