A067201 -id:A067201 - cp's OEIS frontend

A114274 Numbers k such that k^2 + 11 is prime.

0, 6, 24, 30, 36, 54, 90, 96, 114, 120, 126, 144, 150, 180, 186, 204, 210, 234, 246, 270, 300, 324, 366, 390, 444, 456, 486, 504, 510, 564, 636, 654, 666, 684, 690, 720, 774, 780, 834, 846, 864, 930, 936, 954, 960, 984

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), A114271 (i=8), A114272 (i=9), A114273 (i=10), this sequence (i=11), A114275 (i=12).

lst={};k=11;Do[If[PrimeQ[n^2+k], (*Print[n];*)AppendTo[lst, n]], {n, 6!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 26 2008 *)

is(n)=isprime(n^2+11) \\ Charles R Greathouse IV, Jan 21 2015

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

Original entry on oeis.org

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

A108701 Values of n such that n^2-2 and n^2+2 are both prime.

Original entry on oeis.org

3, 9, 15, 21, 33, 117, 237, 273, 303, 309, 387, 429, 441, 447, 513, 561, 573, 609, 807, 897, 1035, 1071, 1113, 1143, 1233, 1239, 1311, 1563, 1611, 1617, 1737, 1749, 1827, 1839, 1953, 2133, 2211, 2283, 2589, 2715, 2721, 2955, 3081, 3093, 3453, 3549, 3555, 3621, 3723, 3807

Offset: 1

John L. Drost, Jun 19 2005

Since x^2 + 2 is divisible by 3 unless x is divisible by 3, all elements are 3 mod 6.

Intersection of A067201 and A028870. - Robert Israel, Sep 11 2014

			21 is on the list since 21^2 - 2 = 439 and 21^2 + 2 = 443 are primes.

David Wells, Prime Numbers, John Wiley and Sons, 2005, p. 219 (article:'Siamese primes')

Nathaniel Johnston, Table of n, a(n) for n = 1..8750
Raymond A. Beauregard and E. R. Suryanarayan, Square-plus-two primes, Mathematical Gazette 85(502) 90-1.

Subsequence of A016945.

Cf. A016945, A028870, A028873, A038599, A067201, A153974.

[n: n in [3..3600 by 6] | IsPrime(n^2-2) and IsPrime(n^2+2)];  // Bruno Berselli, Apr 15 2011

select(n -> isprime(n^2-2) and isprime(n^2+2), [seq(6*i+3,i=0..1000)]); # Robert Israel, Sep 11 2014

Select[Range[5000], PrimeQ[#^2 - 2] && PrimeQ[#^2 + 2] &] (* Alonso del Arte, Sep 11 2014 *)

is(n)=isprime(n^2-2)&&isprime(n^2+2) \\ Charles R Greathouse IV, Jul 02 2013

Terms corrected by Charles R Greathouse IV, Sep 11 2014

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

A086381 Numbers n such that p=n^2+2 and p+2 are primes.

Original entry on oeis.org

1, 3, 15, 33, 45, 57, 117, 147, 243, 255, 303, 375, 423, 447, 453, 477, 573, 753, 837, 897, 903, 1035, 1497, 1905, 2055, 2085, 2193, 2283, 2433, 2487, 2535, 2583, 2757, 2823, 2943, 2955, 3003, 3213, 3285, 3345, 3603, 3657, 3687, 4407, 4575, 4977, 5037, 5043, 5325, 5355, 5367, 5403, 5727

Offset: 1

Zak Seidov, Sep 07 2003

The twin primes are given by A253639 and A085554. Except for the initial term, all a(n)=3 (mod 6). - M. F. Hasler, Jan 16 2015

M. F. Hasler, Table of n, a(n) for n = 1..10000

Cf. A253639, A085554, A059100, A087475.

[n: n in [0..10000]|IsPrime(n^2+2) and IsPrime(n^2+4)] // Vincenzo Librandi, Dec 16 2010

is_A086381(x)=ispseudoprime(x^2+2)&&ispseudoprime(x^2+4)
forstep(x=1,9999,2,is_A086381(x)&&print1(x",")) \\ M. F. Hasler, Jan 16 2015

Intersection of A067201 and A007591. - M. F. Hasler, Jan 19 2015

More terms from Vincenzo Librandi, Dec 16 2010

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

A216976 Numbers k such that k^5+2 is prime.

Original entry on oeis.org

0, 1, 9, 11, 15, 27, 39, 51, 57, 105, 149, 179, 197, 219, 225, 231, 275, 281, 285, 299, 315, 317, 321, 335, 369, 389, 401, 405, 411, 419, 425, 491, 509, 545, 561, 587, 725, 741, 779, 789, 819, 855, 879, 909, 915, 977, 1007, 1019, 1059, 1115, 1145, 1161, 1199

Offset: 1

Michel Lagneau, Sep 21 2012

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A067200, A067201.

select(t->isprime(t^5+2), [$0..10000]); # Robert Israel, Jan 01 2021

lst={}; Do[If[PrimeQ[n^5+2], AppendTo[lst, n]], {n, 0, 10^3}]; lst
Select[Range[0,1200],PrimeQ[#^5+2]&] (* Harvey P. Dale, Jan 30 2025 *)

select(n->isprime(n^5+2),vector(2000,n,n-1)) /* Joerg Arndt, Sep 21 2012 */

A242330 Numbers k such that k^2 + 2 is a semiprime.

Original entry on oeis.org

2, 6, 7, 11, 12, 17, 18, 27, 29, 35, 37, 42, 43, 48, 51, 53, 54, 55, 60, 65, 66, 69, 72, 73, 75, 79, 83, 84, 87, 90, 93, 97, 115, 119, 125, 132, 133, 135, 137, 141, 144, 150, 153, 155, 159, 161, 165, 169, 174, 183, 186, 187, 189, 191, 192, 195, 198

Offset: 1

Vincenzo Librandi, May 14 2014

The semiprimes of this form are: 6, 38, 51, 123, 146, 291, 326, 731, 843, 1227, 1371, 1766, 1851, 2306, 2603, 2811, 2918, 3027, 3602, ....

There are no four consecutive terms in this sequence, that is, a(n) > a(n-3) + 3 (check mod 6). Probably sieve theory can show that this sequence has density 0. - Charles R Greathouse IV, Feb 24 2023

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

Cf. A056899, A067201, A085722, A242331, A242332, A242333.

IsSemiprime:=func; [n: n in [2..200] | IsSemiprime(s) where s is n^2+2];

Mathematica

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

PARI

issemi(n)=forprime(p=2,997,if(n%p==0, return(isprime(n/p)))); bigomega(n)==2 is(n)=issemi(n^2+2) \\ Charles R Greathouse IV, Feb 24 2023

Formula

a(n) > 2n for n > 1. - Charles R Greathouse IV, Feb 24 2023

cp's OEIS Frontend

A114274 Numbers k such that k^2 + 11 is prime.

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

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A108701 Values of n such that n^2-2 and n^2+2 are both 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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A086381 Numbers n such that p=n^2+2 and p+2 are primes.

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

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Magma

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A216976 Numbers k such that k^5+2 is prime.

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