A114270 -id:A114270 - 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

A182238 n^2 + {1,3,7} are primes.

Original entry on oeis.org

2, 4, 10, 74, 146, 256, 440, 470, 584, 920, 1070, 1156, 1324, 1394, 1420, 2080, 2470, 2600, 3326, 3746, 4796, 5996, 6460, 7160, 7466, 8894, 9164, 9554, 9596, 10490, 10970, 11204, 11246, 11336, 11374, 12314, 12386, 13394, 14290, 15586, 16250, 16330, 17060

Offset: 1

Zak Seidov, Apr 20 2012

nonn

Under Schinzel's hypothesis H, this sequence is infinite. - Charles R Greathouse IV, Apr 23 2012

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

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A005574, A049422, A114270.

{ forstep ( n=2, 10^6, 2,
    ns = n * n;
    if ( ! isprime( ns+1 ), next() );
    if ( ! isprime( ns+3 ), next() );
    if ( ! isprime( ns+7 ), next() );
    print1(n, ", ");
); }
/* Joerg Arndt, Apr 22 2012 */

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

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

Zak Seidov, Apr 05 2007

nonn

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

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

Cf. A078402, A114270.

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

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

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

A356110 Numbers k such that k^2 + {1,3,7,13,31} are prime.

Original entry on oeis.org

4, 10, 14290, 43054, 109456, 315410, 352600, 483494, 566296, 685114, 927070, 1106116, 1248796, 1501174, 1997986, 2399204, 2501404, 2553100, 2726840, 2874680, 3291760, 4129394, 4473766, 4794520, 4901144, 6350306, 7444070, 7753456, 7892504, 8009536, 8069540

Offset: 1

Michel Lagneau, Jul 27 2022

nonn

Conjecture: the sequence is infinite.

			4^2 + {1,3,7,13,31} = {17,19,23,29,47} are all prime.

Cf. A005574, A049422, A114270, A113536, A182238, A356109.

q:= k-> andmap(j-> isprime(k^2+j), [1,3,7,13,31]):
select(q, [$0..1000000])[];  # Alois P. Heinz, Jul 27 2022

Select[Range[10^6], AllTrue[#^2 + {1,3,7,13,31}, PrimeQ] &] (* Amiram Eldar, Jul 27 2022 *)

from sympy import isprime
def ok(n): return all(isprime(n*n+i) for i in {1,3,7,13,31})
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jul 27 2022

A356175 Numbers k such that k^2 + {1,3,7,13,163} are prime.

Original entry on oeis.org

2, 4, 10, 14290, 64390, 74554, 83464, 93460, 132304, 238850, 262630, 277630, 300206, 352600, 376190, 404954, 415180, 610340, 806180, 984686, 1025650, 1047050, 1106116, 1382860, 2014624, 2440714, 2525870, 2538344, 2760026, 2826380, 3145600, 3508586, 3715156

Offset: 1

Jean-Marc Rebert, Jul 28 2022

nonn

For 14 <= m <= 999 and k <= A356110(31) = 8069560, the number of sets of primes of the form k^2 + {1,3,7,13,m} is the greatest for m = 163. There are 51 such terms. See b-file.

All terms are 2 or 4 modulo 6.

			2 is a term since 2^2 + {1,3,7,13,163} = {5,7,11,17,167} are all primes.

David A. Corneth, Table of n, a(n) for n = 1..6757 (first 237 terms from Jean-Marc Rebert, terms <= 10^10).

Cf. A356110.

Subsequence of A005574, A049422, A114270, A113536, A080149, A182238 and A356109.

q:= k-> andmap(j-> isprime(k^2+j), [1, 3, 7, 13, 163]):
select(q, [$0..1000000])[];  # Alois P. Heinz, Jul 28 2022

Select[Range[4*10^6], AllTrue[#^2 + {1, 3, 7, 13, 163}, PrimeQ] &] (* Amiram Eldar, Jul 28 2022 *)

is(k)=my(v=[1,3,7,13,163],ok=1);for(i=1,#v,if(!isprime(k^2+v[i]),ok=0;break));ok

from sympy import isprime
def ok(n): return all(isprime(n*n+i) for i in {1,3,7,13,163})
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jul 28 2022

cp's OEIS Frontend

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

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A182238 n^2 + {1,3,7} are primes.

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

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

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

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A356110 Numbers k such that k^2 + {1,3,7,13,31} are prime.

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A356175 Numbers k such that k^2 + {1,3,7,13,163} are prime.

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