A114272 -id:A114272 - cp's OEIS frontend

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

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

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

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

A177173 Numbers n such that n^2 + 3^k is prime for k = 1, 2, 3.

Original entry on oeis.org

2, 10, 38, 52, 350, 542, 1102, 1460, 1522, 1732, 2510, 2642, 2768, 3692, 4592, 4658, 4690, 7238, 8180, 8320, 8960, 11392, 13468, 14920, 15908, 16600, 16832, 17878, 18820, 19100, 21532, 22060, 23240, 23842, 23968, 24622, 26428, 26638, 27170

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 04 2010

nonn

p = n^2 + 3, q = n^2 + 3^2 = p+6, r = n^2 + 3^3 = p+18 to be primes.
Trivially n is not a multiple of 3 and necessarily LSD of such n is e = 0, 2 or 8 as k^2+3^2 is a multiple of 5 for k = 4 or 6.
Note n^2 + m^k prime (k = 1, 2, 3) in case of m = 2 is (n^2+2,n^2+2^2,n^2+2^3) = (p,p+2,p+6): i.e., a "near square" prime triple of the first kind.
Case k=2: q is also a Pythagorean prime (A002144)
n = 350: first case where p = 122503 = prime(i), q and r are consecutive primes (i = 122503), sod(p) = sod(i) = 13, a so-called Honaker prime.
p = prime(i), q, r consecutive primes, (n,i): (350,11524) (542,25517) (1460,157987) (3692,887608) (4592,1335102) (4690,1389018).

			2^2 + 3 = 7 = prime(4), 2^2 + 3^2 = 13 = prime(6), 2^2 + 3^3 = 31 = prime(11), 2 is first term.
10^2 + 3 = 103 = prime(27), 10^2 + 3^2 = 109 = prime(29), 10^2 + 3^3 = 127 = prime(31), 10 is 2nd term.
Curiously k=0: 10^2 + 3^0 = 101 = prime(26), k=4: 10^2 + 3^4 = 181 = prime(42), necessarily LSD for such n is e = 0, k= 5: 10^2 + 3^5 = 7^3, k=6: 10^2 + 3^6 = 829 = prime(145), 10^2 + 3^7 = 2287 = prime(340), 10^2 + 3^8 = 6661 = prime(859)
n = 8180, primes for exponents k = 0, 1, 2, 3 and 4: p=66912403=prime(3946899), q=66912409=prime(3946900), r=66912427=prime(3946902), n^2+3^0=66912401=prime(3946898) and n^2+3^4=66912481=prime(3946905).
n = 8960, primes for exponents k = 1, 2, 3, 4, 5 and 6: p=80281603=prime(4684862), q=80281609=prime(4684863), r=80281627=prime(4684865), n^2+3^4=80281681=prime(4684868), n^2+3^5=80281843=prime(4684877), n^2+3^5=80282329=prime(4684904).

F. Padberg, Zahlentheorie und Arithmetik, Spektrum Akademie Verlag, Heidelberg-Berlin 1999.
M. du Sautoy, Die Musik der Primzahlen: Auf den Spuren des groessten Raetsels der Mathematik, Deutscher Taschenbuch Verlag, 2006.

C. Hooley On nonary cubic forms, Journal für die reine und angewandte Mathematik 386, pp. 32-98, 1988.

Cf. A002144, A049422, A055394, A086380, A114272.

is(n)=isprime(n^2+3) && isprime(n^2+9) && isprime(n^2+27) \\ Charles R Greathouse IV, Jun 13 2017

More terms from R. J. Mathar, Nov 01 2010

cp's OEIS Frontend

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

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

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

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

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

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A177173 Numbers n such that n^2 + 3^k is prime for k = 1, 2, 3.

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