A138375 -id:A138375 - cp's OEIS frontend

A177833 Numbers k such that k^2 - 13 and k^2 + 13 are primes.

4, 12, 18, 72, 84, 114, 198, 354, 378, 588, 612, 618, 864, 912, 948, 1032, 1068, 1134, 1320, 1410, 1428, 1452, 1500, 1830, 1956, 2046, 2058, 2172, 2298, 2448, 2634, 2748, 2844, 2856, 3192, 3246, 3390, 3474, 3846, 3906, 4092, 4182, 4506, 4842, 4884, 4890

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), May 14 2010

nonn

			4^2 - 13 = 3 = prime(2), 4^2 + 13 = 29 = prime(10).
12^2 - 13 = 131 = prime(32), 12^2 + 13 = 157 = prime(37).
948^2 - 13 = 898691 = prime(71194), 948^2 + 13 = prime(71195), first case that they are consecutive primes.

J. Matousek and J. Nesetril, Diskrete Mathematik: eine Entdeckungsreise, Springer-Lehrbuch, 2. Aufl., Berlin, 2007

Zak Seidov, Table of n, a(n) for n = 1..3000

Cf. A138375, A154648, A176969, A176978, A248785.

[n: n in [4..1000]| IsPrime(n^2-13) and IsPrime(n^2+13)]; // Vincenzo Librandi, Nov 30 2010

with(numtheory): A248785:=n->`if`(isprime(n^2-13) and isprime(n^2+13), n, NULL): seq(A248785(n), n=1..10^4); # Wesley Ivan Hurt, Oct 13 2014

Select[Range[2,5000,2],AllTrue[#^2+{13,-13},PrimeQ]&] (* Harvey P. Dale, May 28 2024 *)

More terms from Vincenzo Librandi, May 16 2010

Name edited by Michel Marcus, Nov 25 2024

A228244 Primes of the form k^2 + 17.

Original entry on oeis.org

17, 53, 593, 3617, 4373, 6101, 8117, 11681, 20753, 26261, 30293, 34613, 54773, 63521, 86453, 90017, 101141, 108917, 112913, 116981, 138401, 156833, 176417, 191861, 207953, 213461, 219041, 248021, 278801, 352853, 404513, 419921, 427733, 451601, 518417, 562517

Offset: 1

Michel Marcus, Aug 18 2013

			17 = 0^2 + 17 is prime.
53 = 6^2 + 17 is prime.

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

Cf. A049423, A056899, A056905, A079138, A138362, A138375, A241847, A264790.

[m: n in [0..900] | IsPrime(m) where m is n^2+17]; // Bruno Berselli, Aug 18 2013

Select[Table[n^2 + 17, {n, 0, 900}], PrimeQ] (* Bruno Berselli, Aug 18 2013 *)

PARI
```
isp(n) = isprime(n) && issquare(n-17);
```

a(n) = A241847(A264790(n)). - Elmo R. Oliveira, Apr 21 2025

A243449 Primes of the form n^2 + 14.

Original entry on oeis.org

23, 239, 743, 1103, 2039, 5639, 7583, 8663, 27239, 33503, 38039, 42863, 59063, 81239, 88223, 91823, 119039, 131783, 140639, 164039, 189239, 205223, 245039, 263183, 288383, 328343, 342239, 378239, 393143, 400703, 431663, 439583, 514103, 660983, 710663, 950639

Offset: 1

Vincenzo Librandi, Jun 05 2014

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

Cf. A121250 (associated n).

Cf. primes of the form n^2+k: A144255 (k=1), A056899 (k=2), A049423 (k=3), A005473 (k=4), A056905 (k=5), A056909 (k=6), A079138 (k=7), A138338 (k=8), A138353 (k=9), A138355 (k=10), A138362 (k=11), A138368 (k=12), A138375 (k=13), this sequence (k=14), A243450 (k=15), A243451 (k=16), A228244 (k=17), A174812 (k=42).

[a: n in [0..1000] | IsPrime(a) where a is n^2+14];

Select[Table[n^2 + 14, {n, 0, 2000}], PrimeQ]
Select[Range[1,1001,2]^2+14,PrimeQ] (* Harvey P. Dale, May 30 2023 *)

A176722 Primes of the form k^3 + 13, k >= 0.

Original entry on oeis.org

13, 229, 1013, 1741, 39317, 64013, 74101, 157477, 438989, 551381, 830597, 1906637, 2000389, 4096013, 7077901, 9261013, 10941061, 15625013, 16003021, 21024589, 24897101, 27000013, 69934541, 74088013, 79507013, 93576677, 122023949

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 25 2010

nonn

Necessarily, k = 6 * j or k = 6 * j + 4.

Values of k corresponding to terms of the sequence: 0, 6, 10, 12, 34, 40, 42, 54, 76, 82, 94, 124, 126, 160, 192, 210, 222, 250, 252, 276, 292, 300, 412, 420, 430, 454, 496, 502, 570, 586, 612, 622, 640, 670, 684, 712, 720, 724, 726, 756, 784, 822, 826, 874, 882, 894, 934, 952, 964, 1006, 1056.

			0^3 + 13 = 13 = prime(6) = a(1);
6^3 + 13 = 229 = prime(50) = a(2);
300^3 + 13 = 27000013 = prime(1683067) = a(22).

H. Rademacher, Topics in Analytic Number Theory, Springer-Verlag Berlin, 1973.

Robert Israel, Table of n, a(n) for n = 1..10000
D. R. Heath-Brown and B. Z. Moroz, Primes Represented by Binary Cubic Forms, Proceedings of the London Math. Soc. vol. 84(2), pp. 257-288, 2002; see also.

Cf. A000040, A000578, A037396, A138375, A144953, A154648, A157772, A168147, A168219, A168327.

[a: n in [0..500]|IsPrime(a) where a is n^3+13] // Vincenzo Librandi, Dec 22 2010

select(isprime,[seq(seq((6*j+m)^3+13,m=[0,4]),j=0..1000)]); # Robert Israel, Jun 28 2018

Select[Range[0,1000]^3+13,PrimeQ]  (* Harvey P. Dale, Mar 12 2011 *)

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

A177833 Numbers k such that k^2 - 13 and k^2 + 13 are primes.

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A228244 Primes of the form k^2 + 17.

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A243449 Primes of the form n^2 + 14.

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A176722 Primes of the form k^3 + 13, k >= 0.

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

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