A056902 -id:A056902 - cp's OEIS frontend

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

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

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

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

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

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.

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

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

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 *)

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

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

A056906 Numbers k such that 36*k^2 + 5 is prime.

Original entry on oeis.org

0, 1, 2, 6, 8, 12, 13, 16, 19, 21, 27, 28, 33, 34, 41, 43, 49, 56, 57, 62, 69, 72, 76, 77, 82, 84, 86, 89, 92, 96, 98, 99, 104, 111, 119, 121, 126, 128, 131, 132, 133, 134, 139, 142, 146, 148, 153, 159, 166, 168, 169, 173, 174

Offset: 1

Henry Bottomley, Jul 07 2000

Except for a(1), a(n) is never a multiple of 5.

			a(3)=2 since 36*2^2 + 5 = 149, which is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

This sequence and formula generate all primes of the form k^2+5, i.e., A056905.
Except for the first term, this sequence is a subsequence of A047201.
Cf. A056900, A056902.

[n: n in [0..200]| IsPrime(36*n^2+5)]; // Vincenzo Librandi, Jul 14 2012

Select[Range[0,200],PrimeQ[36#^2+5]&] (* Harvey P. Dale, Jul 25 2011 *)

is(n)=isprime(36*n^2+5) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = sqrt(A056905(n)-5)/6.

Offset corrected by Arkadiusz Wesolowski, Aug 09 2011

A056908 Numbers k such that 36k^2 + 36k + 13 is prime.

Original entry on oeis.org

0, 2, 4, 5, 7, 9, 14, 19, 22, 24, 29, 30, 34, 40, 42, 44, 50, 59, 62, 70, 72, 74, 75, 79, 80, 82, 84, 95, 102, 110, 119, 125, 132, 135, 139, 149, 150, 157, 160, 165, 172, 180, 197, 199, 200, 209, 210, 212, 224, 225, 227, 229, 230, 232, 235, 240, 244, 249

Offset: 1

Henry Bottomley, Jul 07 2000

36*k^2 + 36*k + 13 = (6*k+3)^2 + 4, which is 4 more than a square.

			a(2)=4 since 36*4^2 + 36*4 + 13 = 733, which is prime (as well as being four more than a square).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

This sequence and formula, together with A056907 and its formula, generate all primes of the form k^2+4, i.e., A005473.

Cf. A056900, A056902, A056904, A056906.

[n: n in [0..70]| IsPrime(36*n^2+36*n+13)]; // Vincenzo Librandi, Jul 14 2012

Select[Range[0,700],PrimeQ[36#^2+36#+13]&] (* Vincenzo Librandi, Jul 14 2012 *)

is(n)=isprime(36*n^2+36*n+13) \\ Charles R Greathouse IV, Mar 01 2017

A056907 Numbers k such that 36k^2 + 12k + 5 is prime (sorted by absolute values with negatives before positives).

Original entry on oeis.org

0, -1, 1, 2, -3, -6, 6, -8, -11, 11, 12, 14, -16, 16, 17, 19, -21, -23, -26, 27, -28, 32, -34, -36, 36, -39, 39, -41, 42, 44, -46, 46, -48, -49, 51, 52, -53, -58, 62, 64, 67, -68, -71, 71, -76, 77, 79, 81, -84, -89, 91, 96, -99, -101, 101, 102, -104, -111, 111, -113

Offset: 0

Henry Bottomley, Jul 07 2000

sign

36*k^2 + 12*k + 5 = (6*k+1)^2 + 4, which is four more than a square. Except for a(0), a(n) is never a multiple of 5.

			a(3)=2 since 36*2^2 + 12*2 + 5 = 173 which is prime (as well as being four more than a square).

This sequence and formula, together with A056908 and its formula, generate all primes of the form k^2+4, i.e., A005473. Except for the first term, this sequence is a subsequence of A047201. Cf. A056900, A056902, A056904, A056906.

A056910 Numbers k such that 36k^2 + 12k + 7 is prime (sorted by absolute values with negatives before positives).

Original entry on oeis.org

0, -1, -2, 3, 4, 5, -6, 10, -11, 13, -15, 15, 18, -22, 24, 25, 29, -31, 33, -37, -45, -55, 55, 59, -67, -72, 74, 80, -81, 85, -86, 88, -90, -95, 99, -101, -102, 108, -116, 118, -122, 129, -130, 143, 148, -151, -155, -157, 158, 159, -162, 164, 165

Offset: 0

Henry Bottomley, Jul 07 2000

sign

36*k^2 + 12*k + 7 = (6*k+1)^2 + 6, which is six more than a square.

			a(2)=-2 since 36*(-2)^2 + 12*(-2) + 7 = 127, which is prime (as well as being six more than a square).

This sequence and formula generate all primes of the form k^2+6, i.e., A056909. Except for the first term, none of the a(n) are a multiple of 7 and so the rest of this sequence is a subsequence of A047304. Cf. A056900, A056902, A056904, A056906, A056907, A056908.

a(n) = (-1 +- sqrt(A056909(n) - 6))/6, choosing +- to give an integer result for each n.

cp's OEIS Frontend

A049423 Primes of the form k^2 + 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Magma

Mathematica

PARI

Formula

A056906 Numbers k such that 36*k^2 + 5 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A056908 Numbers k such that 36*k^2 + 36*k + 13 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A056907 Numbers k such that 36*k^2 + 12*k + 5 is prime (sorted by absolute values with negatives before positives).

Views

Author

Keywords

Comments

Examples

Crossrefs

A056910 Numbers k such that 36*k^2 + 12*k + 7 is prime (sorted by absolute values with negatives before positives).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A056908 Numbers k such that 36k^2 + 36k + 13 is prime.

A056907 Numbers k such that 36k^2 + 12k + 5 is prime (sorted by absolute values with negatives before positives).

A056910 Numbers k such that 36k^2 + 12k + 7 is prime (sorted by absolute values with negatives before positives).