A078402 -id:A078402 - cp's OEIS frontend

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

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

A117951 a(n) = n^2 + 5.

Original entry on oeis.org

5, 6, 9, 14, 21, 30, 41, 54, 69, 86, 105, 126, 149, 174, 201, 230, 261, 294, 329, 366, 405, 446, 489, 534, 581, 630, 681, 734, 789, 846, 905, 966, 1029, 1094, 1161, 1230, 1301, 1374, 1449, 1526, 1605, 1686, 1769, 1854, 1941, 2030, 2121, 2214, 2309, 2406, 2505

Offset: 0

Eric W. Weisstein, Apr 04 2006

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Near-Square Prime.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A078402.

Range[0,50]^2+5 (* or *) LinearRecurrence[{3,-3,1},{5,6,9},60] (* Harvey P. Dale, Aug 04 2020 *)

a(n)=n^2+5 \\ Charles R Greathouse IV, Apr 10 2012

[lucas_number1(3,n,-5) for n in range(0, 51)] # Zerinvary Lajos, May 16 2009

a(n) = 2*n + a(n-1) - 1 (with a(0)=5). - Vincenzo Librandi, Nov 13 2010
From Colin Barker, Apr 10 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (5-9*x+6*x^2)/(1-x)^3. (End)
From Amiram Eldar, Jul 13 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(5)*Pi*coth(sqrt(5)*Pi))/10.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(5)*Pi*cosech(sqrt(5)*Pi))/10. (End)
From Amiram Eldar, Feb 05 2024: (Start)
Product_{n>=0} (1 - 1/a(n)) = 2*sinh(2*Pi)/(sqrt(5)*sinh(sqrt(5)*Pi)).
Product_{n>=0} (1 + 1/a(n)) = sqrt(6/5)*sinh(sqrt(6)*Pi)/sinh(sqrt(5)*Pi). (End)
E.g.f.: exp(x)*(5 + x + x^2). - Elmo R. Oliveira, Jan 17 2025

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

A242333 Numbers k such that k^2 + 5 is a semiprime.

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 14, 18, 21, 22, 24, 26, 27, 28, 30, 33, 42, 44, 51, 54, 57, 58, 62, 63, 64, 68, 69, 82, 84, 86, 90, 93, 98, 99, 102, 104, 108, 111, 118, 132, 134, 138, 144, 152, 154, 156, 166, 174, 177, 180, 183, 184, 186, 188, 189, 194, 208, 210, 212, 216

Offset: 1

Vincenzo Librandi, May 14 2014

The semiprimes of this form are: 6, 9, 14, 21, 69, 86, 201, 329, 446, 489, 581, 681, 734, 789, 905, 1094, 1769, 1941, 2606, 2921, 3254, ...

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

Cf. A056905, A078402, A085722, A242330, A242331, A242332.

IsSemiprime:=func; [n: n in [0..300] | IsSemiprime(s) where s is n^2+5];

Mathematica

Select[Range[0, 300], PrimeOmega[#^2 + 5] == 2 &]

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

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Author

Parthasarathy Nambi, Sep 06 2006

Keywords

nonn

Examples

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

Links

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

Crossrefs

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

Programs

Magma

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

Mathematica

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

PARI

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

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

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Author

Parthasarathy Nambi, Sep 14 2006

Keywords

nonn

Examples

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

Links

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

Crossrefs

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

Programs

Magma

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

Mathematica

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

PARI

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

A264790 Numbers k such that k^2 + 17 is prime.

Original entry on oeis.org

0, 6, 24, 60, 66, 78, 90, 108, 144, 162, 174, 186, 234, 252, 294, 300, 318, 330, 336, 342, 372, 396, 420, 438, 456, 462, 468, 498, 528, 594, 636, 648, 654, 672, 720, 750, 798, 804, 834, 858, 888, 924, 930, 966, 984, 990, 1014, 1026, 1032, 1086, 1158, 1194, 1200

Offset: 1

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Author

Ilya Gutkovskiy, Nov 25 2015

Keywords

nonn

Comments

Primes of the form k^2 + 17 have a representation as a sum of 2 squares because they belong to A002144.

All terms are multiple of 6.

Examples

a(3) = 24 because 24^2 + 17 = 593, which is prime.

Links

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

Eric Weisstein's World of Mathematics, Near-Square Prime

Crossrefs

Cf. A228244 (associated primes).

Other sequences of the type "Numbers n such that n^2 + k is prime": A005574 (k=1), A067201 (k=2), A049422 (k=3), A007591 (k=4), A078402 (k=5), A114269 (k=6), A114270 (k=7), A114271 (k=8), A114272 (k=9), A114273 (k=10), A114274 (k=11), A114275 (k=12), A113536 (k=13), A121250 (k=14), A121982 (k=15), A122062 (k=16).

Programs

Magma

[n: n in [0..1200 ] | IsPrime(n^2+17)]; // Vincenzo Librandi, Nov 25 2015

Mathematica

Select[Range[0, 1200], PrimeQ[#^2 + 17] &] (* Michael De Vlieger, Nov 25 2015 *)

PARI

for(n=0, 1e3, if(isprime(n^2+17), print1(n, ", "))) \\ Altug Alkan, Nov 25 2015

Formula

A000005(A241847(a(n))) = 2.

A241847(a(n)) = A228244(n).

Extensions

Edited by Bruno Berselli, Nov 26 2015

A127845 Numbers k such that k^2 + 5 and k^2 + 7 are twin primes.

Original entry on oeis.org

0, 6, 12, 36, 48, 78, 114, 162, 258, 414, 666, 792, 804, 996, 1044, 1206, 1308, 1314, 1356, 1548, 1584, 1632, 1734, 1902, 2106, 2196, 2214, 2346, 2358, 2592, 2634, 2766, 2808, 2868, 2892, 2988, 3072, 3246, 3432, 3516, 3576, 3774, 3894, 3912, 3996, 4038

Offset: 1

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Author

Zak Seidov, Apr 05 2007

Keywords

nonn

Comments

All terms are multiples of 6. Intersection of A114270 and A078402.

Links

Robert Israel, Table of n, a(n) for n = 1..7640

Crossrefs

Cf. A078402, A114270.

Programs

Magma

[n: n in [0..500] | IsPrime(n^2+5) and IsPrime(n^2+7)] // Vincenzo Librandi, Nov 23 2010

Maple

filter:= n -> isprime(n^2+5) and isprime(n^2+7): N:= 10000: # to get all entries <= 6*N A127845:= select(filter, [6*n $ n=0..N]); # Robert Israel, Apr 17 2014

Mathematica

Select[Range[0,5000,6],AllTrue[#^2+{5,7},PrimeQ]&] (* Harvey P. Dale, May 25 2023 *)

Previous Showing 11-19 of 19 results.

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A117951 a(n) = n^2 + 5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

SageMath

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A242333 Numbers k such that k^2 + 5 is a semiprime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A264790 Numbers k such that k^2 + 17 is prime.

Views

Author

Keywords

Comments