A049422 -id:A049422 - cp's OEIS frontend

A114275 Numbers k such that k^2 + 12 is prime.

1, 5, 7, 13, 19, 23, 29, 35, 37, 41, 43, 47, 55, 61, 85, 89, 91, 97, 113, 119, 121, 127, 139, 161, 167, 169, 175, 187, 191, 197, 203, 211, 215, 223, 229, 245, 265, 271, 295, 299, 307, 317, 335, 341, 355, 371, 379, 383, 401, 419, 427, 455, 463, 475, 491, 517, 527

Offset: 1

Zak Seidov, Nov 19 2005

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Other sequences of the type "Numbers k such that k^2 + i is prime": A005574 (i=1), A067201 (i=2), A049422 (i=3), A007591 (i=4), A078402 (i=5), A114269 (i=6), A114270 (i=7), A114271 (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), this sequence (i=12).

Select[Range[1,611,2],PrimeQ[#^2+12]&] (* Harvey P. Dale, Mar 30 2015 *)

is(n)=isprime(n^2+12) \\ Charles R Greathouse IV, Jan 21 2015

A114271 Numbers k such that k^2 + 8 is prime.

Original entry on oeis.org

3, 9, 15, 21, 33, 51, 57, 81, 87, 111, 117, 123, 129, 135, 141, 147, 153, 177, 189, 213, 219, 255, 279, 285, 315, 321, 327, 345, 351, 363, 399, 417, 465, 471, 477, 483, 495, 549, 579, 585, 627, 657, 663, 669, 723, 735, 741, 747, 759, 771, 783, 789, 807, 825

Offset: 1

Zak Seidov, Nov 19 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Other sequences of the type "Numbers k such that k^2 + i is prime": A005574 (i=1), A067201 (i=2), A049422 (i=3), A007591 (i=4), A078402 (i=5), A114269 (i=6), A114270 (i=7), this sequence (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).

a={};Do[If[PrimeQ[n^2+8], AppendTo[a, n]], {n,0,10^3}]; a (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Select[Range[1,901,2],PrimeQ[#^2+8]&] (* Harvey P. Dale, Aug 27 2023 *)

is(n)=isprime(n^2+8) \\ Charles R Greathouse IV, Jan 21 2015

A113536 Numbers k such that k^2 + 13 is prime.

Original entry on oeis.org

0, 2, 4, 10, 12, 16, 18, 28, 40, 42, 44, 46, 60, 68, 72, 82, 84, 88, 94, 108, 110, 114, 116, 122, 126, 142, 144, 152, 158, 180, 192, 194, 198, 200, 220, 222, 264, 266, 268, 282, 284, 296, 298, 332, 336, 340, 354, 378, 380, 418, 420, 430, 434, 446, 464, 466, 486

Offset: 1

Zak Seidov, Jan 13 2006

nonn

			If n=40 then n^2 + 13 = 1613 (prime), so 40 is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Other cases: A005574 k=1, A067201 k=2, A049422 k=3, A007591 k=4, A078402 k=5, A114269-A114275 k=6-12.

With[{k=13}, Select[Range[1000], PrimeQ[ #^2+k]&]]

is(n)=isprime(n^2+13) \\ Charles R Greathouse IV, Feb 17 2017

Edited by R. J. Mathar, Aug 07 2008

A242331 Numbers k such that k^2 + 3 is a semiprime.

Original entry on oeis.org

1, 6, 16, 18, 20, 24, 26, 32, 34, 36, 40, 44, 46, 48, 56, 60, 66, 68, 78, 80, 88, 98, 100, 102, 104, 108, 116, 118, 120, 128, 136, 148, 152, 164, 170, 174, 176, 182, 188, 190, 192, 196, 200, 204, 212, 220, 226, 232, 234, 238, 246, 250, 252, 258, 260, 262, 266

Offset: 1

Vincenzo Librandi, May 14 2014

The semiprimes of this form are: 4, 39, 259, 327, 403, 579, 679, 1027, 1159, 1299, 1603, 1939, 2119, 2307, 3139, 3603, 4359, 4627, ...

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

Cf. A049422, A049423, A085722, A242330, A242332, A242333.

IsSemiprime:=func; [n: n in [0..300] | IsSemiprime(s) where s is n^2+3];

Mathematica

Select[Range[300], PrimeOmega[#^2 + 3] == 2 &]

A121982 Numbers k such that k^2 + 15 is prime.

Original entry on oeis.org

2, 4, 8, 14, 16, 22, 26, 32, 34, 38, 44, 46, 52, 64, 68, 76, 86, 88, 98, 104, 106, 124, 134, 158, 172, 178, 184, 196, 202, 206, 212, 236, 238, 242, 248, 256, 262, 272, 284, 296, 298, 304, 316, 322, 326, 328, 338, 356, 362, 364, 374, 386, 388, 394, 398, 452, 472

Offset: 1

Parthasarathy Nambi, Sep 09 2006

			If k=104 then k^2 + 15 = 10831 (prime).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A005574, A067201, A049422, A007591, A078402, A114269, A114271, A114272, A114273, A114274, A114275, A113536, A121250.

[n: n in [0..6000] | IsPrime(n^2 + 15)]; // Vincenzo Librandi, Nov 13 2010

Select[Range[200], PrimeQ[#1^2 + 15] &] (* G. C. Greubel, Sep 13 2017 *)

is(n)=isprime(n^2+15) \\ Charles R Greathouse IV, Feb 17 2017

A080149 Numbers k such that k^2 + 1 and k^2 + 3 are both prime.

Original entry on oeis.org

2, 4, 10, 14, 74, 94, 130, 134, 146, 160, 230, 256, 326, 340, 350, 406, 430, 440, 470, 584, 634, 686, 700, 704, 784, 860, 920, 986, 1054, 1070, 1156, 1210, 1324, 1340, 1354, 1366, 1394, 1420, 1456, 1460, 1564, 1700, 1784, 1816, 1876, 2006, 2080, 2096, 2174

Offset: 1

T. D. Noe, Jan 30 2003

Hardy and Littlewood conjecture that this sequence is infinite. This sequence is the intersection of A005574 (k such that k^2 + 1 is prime) and A049422 (k such that k^2 + 3 is prime).
From Jacques Tramu, Sep 10 2018: (Start)
a(10000)    =     2473624;  C = 2.91596513
a(100000)   =    35866246;  C = 2.70591741
a(1000000)  =   483764726;  C = 2.53454683
a(2000000)  =  1049178316;  C = 2.49209641
a(3000000)  =  1647417724;  C = 2.46880647
a(4000000)  =  2267125384;  C = 2.45259161
a(5000000)  =  2903162576;  C = 2.44036006
a(6000000)  =  3551848640;  C = 2.43024082
a(7000000)  =  4212006124;  C = 2.42214552
a(8000000)  =  4881390700;  C = 2.41510010
a(9000000)  =  5559542740;  C = 2.40915933
a(10000000) =  6245573750;  C = 2.40405768
a(20000000) = 13393786900;  C = 2.36959294
a(30000000) = 20908970800;  C = 2.35131696
a(40000000) = 28659267134;  C = 2.33835867
a(50000000) = 36590858294;  C = 2.32865934
C is the quotient a(n) / (n * log(n) * log(n)). (End)

			10 is in this sequence because 101 and 103 are both prime.

P. Ribenboim, "The New Book of Prime Number Records," Springer-Verlag, 1996, p. 408.

Zak Seidov, Table of n, a(n) for n = 1..32898 (terms < 10^7, first 1000 terms from T. D. Noe)
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.

Cf. A005574, A049422, A145824.

lst={}; Do[If[PrimeQ[m^2+1]&&PrimeQ[m^2+3], AppendTo[lst, m]], {m, 3000}]; lst
okQ[n_]:=Module[{n2=n^2},PrimeQ[n2+1]&&PrimeQ[n2+3]]; Select[Range[2200], okQ]  (* Harvey P. Dale, Apr 21 2011 *)
Select[Range[2500],AllTrue[#^2+{1,3},PrimeQ]&] (* Harvey P. Dale, Sep 07 2023 *)

isA080149(n) = isprime(n^2+1) && isprime(n^2+3) \\ Michael B. Porter, Mar 22 2010

Conjecture: a(n) is asymptotic to c*n*log(n)^2 with c around 2.9... - Benoit Cloitre, Apr 16 2004

A121250 Numbers n such that n^2 + 14 is prime.

Original entry on oeis.org

3, 15, 27, 33, 45, 75, 87, 93, 165, 183, 195, 207, 243, 285, 297, 303, 345, 363, 375, 405, 435, 453, 495, 513, 537, 573, 585, 615, 627, 633, 657, 663, 717, 813, 843, 975, 1053, 1065, 1083, 1095, 1125, 1137, 1167, 1203, 1287, 1317, 1335, 1353, 1413, 1437, 1455

Offset: 1

Parthasarathy Nambi, Sep 06 2006

nonn

			If n=183 then n^2 + 14 = 33503 (prime).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A005574, A067201, A049422, A007591, A078402, A114269, A114271, A114272, A114273, A114274, A114275, A113536.

[n: n in [0..6000] | IsPrime(n^2+14)] // Vincenzo Librandi, Sep 03 2010

Select[Range[1,1500,2],PrimeQ[#^2+14]&] (* Harvey P. Dale, Aug 20 2011 *)

is(n)=isprime(n^2+14) \\ Charles R Greathouse IV, Feb 20 2017

A182238 n^2 + {1,3,7} are primes.

Original entry on oeis.org

2, 4, 10, 74, 146, 256, 440, 470, 584, 920, 1070, 1156, 1324, 1394, 1420, 2080, 2470, 2600, 3326, 3746, 4796, 5996, 6460, 7160, 7466, 8894, 9164, 9554, 9596, 10490, 10970, 11204, 11246, 11336, 11374, 12314, 12386, 13394, 14290, 15586, 16250, 16330, 17060

Offset: 1

Zak Seidov, Apr 20 2012

nonn

Under Schinzel's hypothesis H, this sequence is infinite. - Charles R Greathouse IV, Apr 23 2012

			2^2+{1,3,7}= {5,7,11} all prime, 4^2+{1,3,7}= {17,19,23} all prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A005574, A049422, A114270.

{ forstep ( n=2, 10^6, 2,
    ns = n * n;
    if ( ! isprime( ns+1 ), next() );
    if ( ! isprime( ns+3 ), next() );
    if ( ! isprime( ns+7 ), next() );
    print1(n, ", ");
); }
/* Joerg Arndt, Apr 22 2012 */

A153975 Values of n such that n^2-3 and n^2+3 are both prime.

Original entry on oeis.org

4, 8, 10, 14, 64, 92, 112, 140, 146, 172, 218, 298, 304, 322, 326, 340, 350, 356, 416, 440, 470, 508, 554, 560, 580, 626, 634, 652, 668, 686, 694, 704, 728, 736, 746, 770, 806, 818, 868, 892, 920, 1054, 1082, 1102, 1130, 1156, 1196, 1256, 1264, 1378, 1418

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Intersection of A028873 and A049422. - Zak Seidov, Oct 12 2014

			4^2 - 3 = 13 and 4^2 + 3 = 19 are both primes, so 4 is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A028870, A028873, A038599, A049422, A108701, A153974.

[n: n in [1..1400] | IsPrime(n^2-3) and IsPrime(n^2+3)]; // Vincenzo Librandi, Oct 12 2014

Select[Range[1500], PrimeQ[#^2 - 3] && PrimeQ[#^2 + 3] &] (* Vincenzo Librandi, Oct 12 2014 *)

is(n) = isprime(n^2-3) && isprime(n^2+3); \\ Altug Alkan, Sep 01 2016

Incorrect term 0 removed and Mma edited by Zak Seidov, Oct 12 2014

A122062 Numbers k such that k^2 + 16 is prime.

Original entry on oeis.org

1, 5, 9, 11, 15, 21, 25, 29, 31, 41, 49, 51, 55, 65, 75, 79, 81, 89, 91, 95, 99, 109, 115, 119, 121, 125, 129, 151, 165, 179, 191, 211, 219, 221, 229, 231, 245, 249, 265, 275, 281, 289, 291, 295, 299, 301, 311, 315, 335, 351, 355, 361, 365, 369, 381, 389, 391

Offset: 1

Parthasarathy Nambi, Sep 14 2006

nonn

			If k=99 then k^2 + 16 = 9817 (prime).

Daniel Starodubtsev, Table of n, a(n) for n = 1..5000

Cf. A005574, A067201, A049422, A007591, A078402, A114269, A114271, A114272, A114273, A114274, A114275, A113536, A121250, A121982.

[n: n in [0..6000] | IsPrime(n^2 + 16)]; // Vincenzo Librandi, Nov 13 2010

Select[Range[1,501,2],PrimeQ[16+#^2]&] (* Harvey P. Dale, Dec 04 2018 *)

is(n)=isprime(n^2+16) \\ Charles R Greathouse IV, Jun 06 2017

cp's OEIS Frontend

A114275 Numbers k such that k^2 + 12 is prime.

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A114271 Numbers k such that k^2 + 8 is prime.

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A113536 Numbers k such that k^2 + 13 is prime.

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A242331 Numbers k such that k^2 + 3 is a semiprime.

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A121982 Numbers k such that k^2 + 15 is prime.

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A080149 Numbers k such that k^2 + 1 and k^2 + 3 are both prime.

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A121250 Numbers n such that n^2 + 14 is prime.

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A182238 n^2 + {1,3,7} are primes.

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