A049422 -id:A049422 - cp's OEIS frontend

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

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

A153262 Squares such that square+-3=primes.

Original entry on oeis.org

16, 64, 100, 196, 4096, 8464, 12544, 19600, 21316, 29584, 47524, 88804, 92416, 103684, 106276, 115600, 122500, 126736, 173056, 193600, 220900, 258064, 306916, 313600, 336400, 391876, 401956, 425104, 446224, 470596, 481636, 495616, 529984

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 22 2008

nonn

16-3=13,16+3=19,primes; 64-3=61,64+3=67,primes;...

The squared members of the intersection of A028873 and A049422. [From R. J. Mathar, Jan 03 2009]

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

Cf. A144938

lst={};Do[p=n^2;If[PrimeQ[p-3]&&PrimeQ[p+3],AppendTo[lst,p]],{n,7!}];lst
Select[Range[750]^2,And@@PrimeQ[#+{3,-3}]&] (* Harvey P. Dale, Dec 19 2012 *)

A155962 Numbers n with property that 3(2n)^2+1 and 1(2n)^2+3 are primes.

Original entry on oeis.org

1, 4, 11, 32, 56, 73, 80, 109, 122, 143, 158, 175, 182, 217, 256, 262, 280, 284, 290, 308, 343, 347, 403, 431, 434, 437, 535, 581, 598, 619, 655, 665, 928, 973, 980, 1018, 1036, 1046, 1096, 1120, 1159, 1207, 1222, 1235, 1267, 1382, 1393, 1439, 1460, 1463, 1501

Offset: 1

Zak Seidov, Jan 31 2009

nonn

2*A155962 is intersection of A049422 and A111051.

			n=1, {3*(2n)^2+1, 1*(2n)^2+3}={13,7};
n=4, {3*(2n)^2+1, 1*(2n)^2+3}={193,67};
n=11, {3*(2n)^2+1, 1*(2n)^2+3}={1453,487};
n=32, {3*(2n)^2+1,1*(2n)^2+3}={12289,4099}.
Resulting primes are congruent to 1 mod 3.

Zak Seidov, Table of n,a (n) for n=1..1000

Cf. A049422, A111051.

Select[Range[1600],AllTrue[{3(2#)^2+1,(2#)^2+3},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 30 2016 *)

All the terms in the b-file had to be divided by 2. Corrected by N. J. A. Sloane, Aug 31 2009.

A329102 Numbers k such that both k^2 + 3 and 2^k + 3 are primes.

Original entry on oeis.org

2, 4, 28, 784

Offset: 1

Alex Ratushnyak, Nov 04 2019

Intersection of A057732 and A049422.
a(5) > 20000. - Tyler NeSmith, May 15 2021
a(5) > 2205444, using A057732. - Michael S. Branicky, Feb 16 2024

Cf. A000040, A057732, A049422, A329103.

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

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

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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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A153262 Squares such that square+-3=primes.

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A155962 Numbers n with property that 3*(2n)^2+1 and 1*(2n)^2+3 are primes.

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A329102 Numbers k such that both k^2 + 3 and 2^k + 3 are 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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A177173 Numbers n such that n^2 + 3^k is prime for k = 1, 2, 3.

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A155962 Numbers n with property that 3(2n)^2+1 and 1(2n)^2+3 are primes.