A114273 -id:A114273 - cp's OEIS frontend

A005574 Numbers k such that k^2 + 1 is prime.

1, 2, 4, 6, 10, 14, 16, 20, 24, 26, 36, 40, 54, 56, 66, 74, 84, 90, 94, 110, 116, 120, 124, 126, 130, 134, 146, 150, 156, 160, 170, 176, 180, 184, 204, 206, 210, 224, 230, 236, 240, 250, 256, 260, 264, 270, 280, 284, 300, 306, 314, 326, 340, 350, 384, 386, 396

Offset: 1

N. J. A. Sloane

Hardy and Littlewood conjectured that the asymptotic number of elements in this sequence not exceeding n is approximately c*sqrt(n)/log(n) for some constant c. - Stefan Steinerberger, Apr 06 2006
Also, nonnegative integers such that a(n)+i is a Gaussian prime. - Maciej Ireneusz Wilczynski, May 30 2011
Apparently Goldbach conjectured that any a > 1 from this sequence can be written as a=b+c where b and c are in this sequence (Lemmermeyer link below). - Jeppe Stig Nielsen, Oct 14 2015
No term > 2 can be both in this sequence and in A001105 because of the Aurifeuillean factorization (2*k^2)^2 + 1 = (2*k^2 - 2*k + 1) * (2*k^2 + 2*k + 1). - Jeppe Stig Nielsen, Aug 04 2019

Harvey Dubner, "Generalized Fermat primes", J. Recreational Math., 18 (1985): 279-280.
R. K. Guy, "Unsolved Problems in Number Theory", 3rd edition, A2.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, p. 15, Thm. 17.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
H. Dubner, Generalized Fermat primes, J. Recreational Math. 18.4 (1985-1986), 279. (Annotated scanned copy)
F. Ellermann, Primes of the form (m^2)+1 up to 10^6.
L. Euler, Lettre CXLIX (to Goldbach), 1752.
L. Euler, De numeris primis valde magnis, Novi Commentarii academiae scientiarum Petropolitanae 9 (1764), pp. 99-153. See pp. 123-125.
R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy).
F. Lemmermeyer, Primes of the form a^2+1, Math Overflow question (2010).
Eric Weisstein's World of Mathematics, Landau's Problems.
Eric Weisstein's World of Mathematics, Power.
Eric Weisstein's World of Mathematics, Near-Square Prime.
Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Cf. A002522, A001912, A002496, A062325, A090693, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.
Other sequences of the type "Numbers k such that k^2 + i is prime": this sequence (i=1), A067201 (i=2), A049422 (i=3), A007591 (i=4), A078402 (i=5), A114269 (i=6), A114270 (i=7), A114271 (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).
Cf. A010051, A259645, A295405 (characteristic function).

a005574 n = a005574_list !! (n-1)
a005574_list = filter ((== 1) . a010051' . (+ 1) . (^ 2)) [0..]
-- Reinhard Zumkeller, Jul 03 2015

[n: n in [0..400] | IsPrime(n^2+1)]; // Vincenzo Librandi, Nov 18 2010

Select[Range[350], PrimeQ[ #^2 + 1] &] (* Stefan Steinerberger, Apr 06 2006 *)
Join[{1},2Flatten[Position[PrimeQ[Table[x^2+1,{x,2,1000,2}]],True]]]  (* Fred Patrick Doty, Aug 18 2017 *)

isA005574(n) = isprime(n^2+1) \\ Michael B. Porter, Mar 20 2010

for(n=1, 1e3, if(isprime(n^2 + 1), print1(n, ", "))) \\ Altug Alkan, Oct 14 2015

from sympy import isprime; [print(n, end = ', ') for n in range(1, 400) if isprime(n*n+1)] # Ya-Ping Lu, Apr 23 2025

a(n) = A090693(n) - 1.

a(n) = 2*A001912(n-1) for n > 1. - Jeppe Stig Nielsen, Aug 04 2019

A067201 Numbers k such that k^2 + 2 is prime.

Original entry on oeis.org

0, 1, 3, 9, 15, 21, 33, 39, 45, 57, 81, 99, 105, 111, 117, 123, 147, 171, 219, 225, 237, 243, 249, 255, 273, 297, 303, 309, 321, 345, 351, 363, 369, 375, 387, 417, 423, 429, 441, 447, 453, 477, 501, 513, 549, 555, 561, 573, 603, 609, 651, 675, 681, 699, 711, 753

Offset: 1

Benoit Cloitre, Feb 19 2002

nonn

All terms > 1 are divisible by 3. - Robert Israel, Sep 05 2014

T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein's World of Mathematics, Near-Square Prime

Equals 6*A056900(n-2) + 3, n>1.
Cf. A106571, A040976.
Other sequences of the type "Numbers k such that k^2 + i is prime": A005574 (i=1), this sequence (i=2), A049422 (i=3), A007591 (i=4), A078402 (i=5), A114269 (i=6), A114270 (i=7), A114271 (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).

select(t -> isprime(t^2+2), [0,1,seq(3*i,i=1..1000)]); # Robert Israel, Sep 05 2014

lst={};Do[If[PrimeQ[n^2+2], AppendTo[lst, n]], {n, 3*10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
Join[{0, 1}, Select[Range[3, 1000, 6], PrimeQ[#^2 + 2] &]] (* Zak Seidov, Jan 30 2014 *)

select(n -> isprime(n^2+2),[1..500]) \\ Edward Jiang, Sep 05 2014

A049422 Numbers k such that k^2 + 3 is prime.

Original entry on oeis.org

0, 2, 4, 8, 10, 14, 22, 28, 38, 50, 52, 62, 64, 70, 74, 76, 92, 94, 106, 112, 122, 130, 134, 140, 146, 154, 158, 160, 172, 178, 218, 230, 242, 244, 248, 256, 274, 286, 298, 304, 316, 322, 326, 340, 350, 356, 364, 368, 398, 406, 416, 424, 430, 434, 440, 458, 470

Offset: 1

Paul Jobling (paul.jobling(AT)whitecross.com)

			4^2 + 3 = 19, which is prime.

Zak Seidov, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Near-Square Prime

Other sequences of the type "Numbers k such that k^2 + i is prime": A005574 (i=1), A067201 (i=2), this sequence (i=3), A007591 (i=4), A078402 (i=5), A114269 (i=6), A114270 (i=7), A114271 (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).

lst={};Do[If[PrimeQ[n^2+3], Print[n];AppendTo[lst, n]], {n, 10^5}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)

isA049422(n) = isprime(n^2+3) \\ Michael B. Porter, Mar 19 2010

A007591 Numbers k such that k^2 + 4 is prime.

Original entry on oeis.org

1, 3, 5, 7, 13, 15, 17, 27, 33, 35, 37, 45, 47, 57, 65, 67, 73, 85, 87, 95, 97, 103, 115, 117, 125, 135, 137, 147, 155, 163, 167, 177, 183, 193, 203, 207, 215, 217, 233, 235, 243, 245, 253, 255, 265, 267, 275, 277, 287, 293, 303, 307, 313, 317, 347, 357, 373, 375

Offset: 1

N. J. A. Sloane, R. K. Guy

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Near-Square Prime

Other sequences of the type "Numbers k such that k^2 + i is prime": A005574 (i=1), A067201 (i=2), A049422 (i=3), this sequence (i=4), A078402 (i=5), A114269 (i=6), A114270 (i=7), A114271 (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).

[n: n in [0..400] | IsPrime(n^2+4)]; // Vincenzo Librandi, Nov 18 2010

lst={};Do[If[PrimeQ[n^2+4], Print[n];AppendTo[lst, n]], {n, 10^5}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
Select[Range[0, 400], PrimeQ[#^2 + 4] &] (* Vincenzo Librandi, Sep 25 2012 *)

select(n->isprime(n^2+4),vector(500,n,2*n-1)) \\ Charles R Greathouse IV, Sep 25 2012

A078402 Numbers k such that k^2 + 5 is prime.

Original entry on oeis.org

0, 6, 12, 36, 48, 72, 78, 96, 114, 126, 162, 168, 198, 204, 246, 258, 294, 336, 342, 372, 414, 432, 456, 462, 492, 504, 516, 534, 552, 576, 588, 594, 624, 666, 714, 726, 756, 768, 786, 792, 798, 804, 834, 852, 876, 888, 918, 954, 996

Offset: 1

Cino Hilliard, Dec 26 2002

The sum of the reciprocals of the primes generated from these indices converges to 0.2332142.. The sum of the reciprocals of these indices cannot be computed.

All terms are divisible by 6. - Zak Seidov, Dec 27 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Near-Square Prime

For the primes see A056905(n).

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), this sequence (i=5), A114269 (i=6), A114270 (i=7), A114271 (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).

[n: n in [0..1000]| IsPrime(n^2+5)]; // Vincenzo Librandi, Jul 15 2012

Select[Range[0,1000],PrimeQ[#^2+5]&] (* Vincenzo Librandi, Jul 13 2012 *)

for(n=0,10^4,q=n^2+5;if(isprime(q),print1(n,", ")));

a(n) = 6 * A056906(n).

Offset corrected by Arkadiusz Wesolowski, Aug 09 2011

A114270 Numbers k such that k^2 + 7 is prime.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 16, 18, 22, 26, 30, 32, 34, 36, 38, 40, 48, 52, 58, 60, 62, 66, 74, 76, 78, 100, 106, 110, 114, 116, 118, 120, 122, 124, 132, 136, 138, 144, 146, 148, 158, 162, 164, 176, 184, 186, 190, 192, 194, 206, 208, 216, 220, 228, 232, 248, 250, 256, 258

Offset: 1

Zak Seidov, Nov 19 2005

nonn

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

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), this sequence (i=7), A114271 (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).

Select[Range[0,1000],PrimeQ[#^2+7]&] (* Vladimir Joseph Stephan Orlovsky, Feb 27 2011 *)

is(n)=isprime(n^2+7) \\ Charles R Greathouse IV, Jan 21 2015

A114269 Numbers k such that k^2 + 6 is prime.

Original entry on oeis.org

1, 5, 11, 19, 25, 31, 35, 61, 65, 79, 89, 91, 109, 131, 145, 151, 175, 185, 199, 221, 269, 329, 331, 355, 401, 431, 445, 481, 485, 511, 515, 529, 539, 569, 595, 605, 611, 649, 695, 709, 731, 775, 779, 859, 889, 905, 929, 941, 949, 955, 971, 985, 991

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), A114270 (i=7), A114271 (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).

With[{k=6}, Select[Range[1000], PrimeQ[ #^2+k]&]]

is(n)=isprime(n^2+6) \\ Charles R Greathouse IV, Jan 21 2015

A114272 Numbers k such that k^2 + 9 is prime.

Original entry on oeis.org

2, 8, 10, 20, 32, 38, 40, 52, 58, 62, 70, 82, 88, 98, 100, 110, 112, 118, 140, 142, 160, 170, 188, 190, 200, 202, 212, 218, 220, 242, 298, 308, 320, 332, 350, 358, 368, 380, 382, 400, 410, 412, 422, 448, 472, 482, 490, 502, 512, 530, 538, 542, 568, 572, 578

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), A114271 (i=8), this sequence (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).

lst={};Do[If[PrimeQ[n^2+9], AppendTo[lst, n]], {n, 600}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 26 2008 *)
Select[Range[2,600,2],PrimeQ[#^2+9]&] (* Harvey P. Dale, Feb 28 2023 *)

is(n)=isprime(n^2+9) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = 2 * A002970(n). - Michel Marcus, Jan 20 2015

A114274 Numbers k such that k^2 + 11 is prime.

Original entry on oeis.org

0, 6, 24, 30, 36, 54, 90, 96, 114, 120, 126, 144, 150, 180, 186, 204, 210, 234, 246, 270, 300, 324, 366, 390, 444, 456, 486, 504, 510, 564, 636, 654, 666, 684, 690, 720, 774, 780, 834, 846, 864, 930, 936, 954, 960, 984

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), A114271 (i=8), A114272 (i=9), A114273 (i=10), this sequence (i=11), A114275 (i=12).

lst={};k=11;Do[If[PrimeQ[n^2+k], (*Print[n];*)AppendTo[lst, n]], {n, 6!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 26 2008 *)

is(n)=isprime(n^2+11) \\ Charles R Greathouse IV, Jan 21 2015

A114275 Numbers k such that k^2 + 12 is prime.

Original entry on oeis.org

1, 5, 7, 13, 19, 23, 29, 35, 37, 41, 43, 47, 55, 61, 85, 89, 91, 97, 113, 119, 121, 127, 139, 161, 167, 169, 175, 187, 191, 197, 203, 211, 215, 223, 229, 245, 265, 271, 295, 299, 307, 317, 335, 341, 355, 371, 379, 383, 401, 419, 427, 455, 463, 475, 491, 517, 527

Offset: 1

Zak Seidov, Nov 19 2005

nonn

Harvey P. Dale, 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), A114271 (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), this sequence (i=12).

Select[Range[1,611,2],PrimeQ[#^2+12]&] (* Harvey P. Dale, Mar 30 2015 *)

is(n)=isprime(n^2+12) \\ Charles R Greathouse IV, Jan 21 2015

cp's OEIS Frontend

A005574 Numbers k such that k^2 + 1 is prime.

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A067201 Numbers k such that k^2 + 2 is prime.

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A049422 Numbers k such that k^2 + 3 is prime.

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A007591 Numbers k such that k^2 + 4 is prime.

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A078402 Numbers k such that k^2 + 5 is prime.

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A114270 Numbers k such that k^2 + 7 is prime.

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A114269 Numbers k such that k^2 + 6 is prime.

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