A113536 -id:A113536 - cp's OEIS frontend

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

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

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

A138375 Primes of the form k^2 + 13.

Original entry on oeis.org

13, 17, 29, 113, 157, 269, 337, 797, 1613, 1777, 1949, 2129, 3613, 4637, 5197, 6737, 7069, 7757, 8849, 11677, 12113, 13009, 13469, 14897, 15889, 20177, 20749, 23117, 24977, 32413, 36877, 37649, 39217, 40013, 48413, 49297, 69709, 70769, 71837, 79537, 80669, 87629

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 07 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..4500

Cf. A113536, A241749.

[a: n in [0..700] | IsPrime(a) where a is n^2+13]; // Vincenzo Librandi, Nov 30 2011

Intersection[Table[n^2+13,{n,0,10^2}],Prime[Range[9*10^3]]] ...or... For[i=13,i<=13,a={};Do[If[PrimeQ[n^2+i],AppendTo[a,n^2+i]],{n,0,100}];Print["n^2+",i,",",a];i++ ]
Select[Table[n^2+13,{n,0,7000}],PrimeQ] (* Vincenzo Librandi, Nov 30 2011 *)

a(n) = A241749(A113536(n)). - Elmo R. Oliveira, Apr 20 2025

A176969 Numbers n such that n^2 + 13^2 is prime.

Original entry on oeis.org

2, 8, 10, 12, 20, 22, 28, 30, 32, 38, 42, 48, 58, 60, 62, 68, 80, 90, 100, 108, 110, 112, 122, 128, 138, 142, 148, 150, 168, 172, 180, 190, 198, 200, 202, 210, 228, 230, 232, 238, 242, 248, 258, 262, 268, 280, 282, 302, 310, 318, 340, 342, 360, 362, 368, 378

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 29 2010

nonn

The n^2 + d conjecture is a famous and still unsolved problem.
It states that there exist an infinite number of primes whose values are of the form n^2 + d for some integer n.
This is case d = 13^2.

			2^2 + 13^2 = 173 = prime(40), 2 is first term.
12^2 + 13^2 = 313 = prime(65) = palprime(11), 12 is 4th term.
310^2 + 13^2 = 96269 = prime(9274) = palprime(106), 310 the 49th term.

J. Matousek, J. Nesetril: Diskrete Mathematik: eine Entdeckungsreise, Springer-Lehrbuch, 2. Aufl., Berlin, 2007
M. du Sautoy: Die Musik der Primzahlen: Auf den Spuren des groessten Raetsels der Mathematik, Deutscher Taschenbuch Verlag, 2006

Cf. A000040, A000290, A056899, A113536, A176371.

isok(n) = isprime(n^2 + 13^2) \\ Michel Marcus, Jun 28 2013

A176978 Numbers n such that n^2 + 13 and n^2 + 13^2 are primes.

Original entry on oeis.org

2, 10, 12, 28, 42, 60, 68, 108, 110, 122, 142, 180, 198, 200, 268, 282, 340, 378, 380, 418, 430, 488, 500, 502, 530, 612, 742, 788, 802, 838, 840, 912, 942, 948, 952, 1010, 1080, 1102, 1148, 1232, 1270, 1428, 1452, 1472, 1502, 1522, 1538, 1720, 1778, 1830

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 30 2010

nonn

Numbers are subsequence of A113536 and A176969

See comments and references of A176969

			2^2 + 13 = 17 = prime(7), 2^2 + 13^2 = 173 = prime(40), 2 is first term.

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

Cf. A000040, A000290, A056899, A113536, A176371, A176969.

Select[Range[2000],AllTrue[#^2+{13,13^2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 28 2015 *)

isok(n) = isprime(n^2 + 13) && isprime(n^2 + 13^2); \\ Michel Marcus, Aug 27 2013

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

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

Ilya Gutkovskiy, Nov 25 2015

nonn

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.

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

Daniel Starodubtsev, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Near-Square Prime

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).

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

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

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

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

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

Edited by Bruno Berselli, Nov 26 2015

A356109 Numbers k such that k^2 + {1,3,7,13} are prime.

Original entry on oeis.org

2, 4, 10, 5996, 8894, 11204, 14290, 23110, 30866, 37594, 43054, 64390, 74554, 83464, 93460, 109456, 111940, 132304, 151904, 184706, 238850, 262630, 265990, 277630, 300206, 315410, 352600, 355450, 376190, 404954, 415180, 462830, 483494, 512354, 512704, 566296

Offset: 1

Michel Lagneau, Jul 27 2022

nonn

Conjecture: the sequence is infinite.

			2^2 + {1,3,7,13} = {5,7,11,17} all prime.
4^2 + {1,3,7,13} = {17,19,23,29} all prime.

Intersection of A005574, A049422, A114270, A113536.

Subsequence of A182238.

q:= k-> andmap(j-> isprime(k^2+j), [1,3,7,13]):
select(q, [$0..1000000])[];  # Alois P. Heinz, Jul 27 2022

Select[Range[500000], AllTrue[#^2 + {1,3,7,13}, PrimeQ] &] (* Amiram Eldar, Jul 27 2022 *)

from sympy import isprime
def ok(n): return all(isprime(n*n+i) for i in {1,3,7,13})
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jul 27 2022

A113537 Numbers k such that k^2 + 11 and k^2 + 13 are primes.

Original entry on oeis.org

114, 126, 144, 180, 486, 684, 864, 1080, 1176, 1866, 1956, 2166, 2226, 2454, 2634, 2880, 3090, 3474, 4176, 4314, 5124, 5586, 5730, 5820, 6030, 6276, 6324, 6456, 6624, 6846, 6954, 7110, 7830, 7914, 8166, 8556, 8724, 9054, 9180, 9576, 9786, 9816

Offset: 1

Zak Seidov, Jan 13 2006

G. C. Greubel, Table of n, a(n) for n = 1..2500

Intersection of A114274 and A113536.

Select[Range[10000], PrimeQ[ #^2+11]&&PrimeQ[ #^2+13]&]

A178504 Numbers n such that n^2 + 13 is an emirp.

Original entry on oeis.org

2, 10, 12, 18, 44, 60, 88, 108, 110, 114, 116, 122, 192, 198, 282, 380, 446, 574, 588, 604, 612, 618, 838, 840, 864, 970, 1032, 1068, 1104, 1148, 1186, 1228, 1258, 1314, 1368, 1384, 1390, 1412, 1754, 1888, 1894, 1930, 2658, 2660, 2728, 2784, 2804

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), May 29 2010

A decimal emirp/mirp ("prime" / (German) "prim", spelled backwards) is defined as a prime number p whose reversal R(p) is also prime, but which is not a palindromic prime.

			2^2 + 13 = 17 = prime(7), 71 = prime(20), so 2 is in the sequence.
10^2 + 13 = 113 = prime(30), 311 = prime(64), so 10 is in the sequence.
28^2 + 13 = 797, which is a palindromic prime, so 28 is not in the sequence.

W. W. R. Ball, H. S. M. Coxeter: Mathematical Recreations and Essays, 13th edition, Dover Publications, 2010
H. Steinhaus: Kaleidoskop der Mathematik, VEB Dt. Verl. d. Wissenschaften, Berlin, 1959

Vincenzo Librandi, Table of n, a(n) for n = 1..8000

Subsequence of A113536.

Cf. A138375, A176978, A178044.

fQ[n_] := If[ PrimeQ[n^2 + 13], Block[{id = IntegerDigits[n^2 + 13]}, rid = Reverse@ id; PrimeQ@ FromDigits@ rid && rid != id]]; Select[ Range@ 3000, fQ] (* Robert G. Wilson v, Jul 26 2010 *)

More terms from Robert G. Wilson v, Jul 26 2010

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A138375 Primes of the form k^2 + 13.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A176969 Numbers n such that n^2 + 13^2 is prime.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

PARI

A176978 Numbers n such that n^2 + 13 and n^2 + 13^2 are primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

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

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A356109 Numbers k such that k^2 + {1,3,7,13} are prime.