cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A049423 Primes of the form k^2 + 3.

Original entry on oeis.org

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

Views

Author

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

Keywords

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

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

A097698 Numbers k such that both 4*k^2 - 3 and 4*k^2 + 3 are primes.

Original entry on oeis.org

2, 4, 5, 7, 32, 46, 56, 70, 73, 86, 109, 149, 152, 161, 163, 170, 175, 178, 208, 220, 235, 254, 277, 280, 290, 313, 317, 326, 334, 343, 347, 352, 364, 368, 373, 385, 403, 409, 434, 446, 460, 527, 541, 551, 565, 578, 598, 628, 632, 689, 709, 710, 737, 761, 812
Offset: 1

Views

Author

Carl R. White, Aug 20 2004

Keywords

Links

Crossrefs

Cf. A083022, A097697, A153975.

Programs

  • Magma
    [n: n in [1..1000] |IsPrime(4*n^2-3) and IsPrime(4*n^2+3)]; // Vincenzo Librandi, Nov 16 2010
  • Mathematica
    Select[Range[0,7! ],PrimeQ[ #^2-3]&&PrimeQ[ #^2+3]&]/2 (* Vladimir Joseph Stephan Orlovsky, Apr 23 2010 *)
Select[Range[1000],AllTrue[4#^2+{3,-3},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 08 2019 *)
  • PARI
    is(n)=isprime(4*n^2-3) && isprime(4*n^2+3) \\ Charles R Greathouse IV, Sep 27 2016

Formula

a(n) = A153975(n) / 2. - Vladimir Joseph Stephan Orlovsky, Apr 23 2010

A121825 Duplicate of A049423.

Original entry on oeis.org

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, 92419
Offset: 1

Views

Author

Jonathan Vos Post, Aug 27 2006

Keywords

Comments

See also A121326 (Primes of the form 4*k^2 + 1); see also A049423 (Primes of the form k^2 + 3). For the primes of the form 4*k^2 + 3, the corresponding values of k are 1, 2, 4, 5, 7, 11, 14, 19, 25, 26, 31, 32, 35, 37, 38, 46, 47, 53, 56, 61, 65, 67, 70, 73.

Crossrefs

Cf. A000040, A002496, A049423, A121326.

Programs

  • Magma
    [ a: n in [0..200] | IsPrime(a) where a is 4*n^2+3 ]; // Vincenzo Librandi, Dec 22 2010

Formula

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

Extensions

Terms after 21319 added by R. J. Mathar, Aug 07 2008
3 added by Vincenzo Librandi, Dec 22 2010
Showing 1-3 of 3 results.

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