A067201 -id:A067201 - 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

A056899 Primes of the form k^2 + 2.

Original entry on oeis.org

2, 3, 11, 83, 227, 443, 1091, 1523, 2027, 3251, 6563, 9803, 11027, 12323, 13691, 15131, 21611, 29243, 47963, 50627, 56171, 59051, 62003, 65027, 74531, 88211, 91811, 95483, 103043, 119027, 123203, 131771, 136163, 140627, 149771, 173891

Offset: 1

Henry Bottomley, Jul 05 2000

nonn

Also, primes of the form k^2 - 2k + 3.
Note that all terms after the first two are equal to 11 modulo 72 and that (a(n)-11)/72 is a triangular number, since they have to be 2 more than the square of an odd multiple of 3 to be prime, and if k = 6*m+3 then a(n) = k^2 + 2 = 72*m*(m+1)/2 + 11.
The quotient cycle length is 2 in the continued fraction expansion of sqrt(p) for these primes. E.g.: cfrac(sqrt(6563),6) = 81+1/(81+1/(162+1/(81+1/(162+1/(81+1/(162+`...`)))))). - Labos Elemer, Feb 22 2001
Primes in A059100; except for a(2)=3 a subsequence of A007491 and congruent to 2 modulo 9. For n>2, a(n)=11 (mod 72). - M. F. Hasler, Apr 05 2009

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988.
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997.

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

Intersection of A146327 and A000040; intersection of A059100 and A000040.

Cf. A002496.

[n: n in PrimesUpTo(175000) | IsSquare(n-2)];  // Bruno Berselli, Apr 05 2011

[ a: n in [0..450] | IsPrime(a) where a is n^2 +2 ]; // Vincenzo Librandi, Apr 06 2011

select(isprime, [seq(t^2+2, t = 0..1000)]); # Robert Israel, Sep 03 2015

Select[ Range[0, 500]^2 + 2, PrimeQ] (* Robert G. Wilson v, Sep 03 2015 *)

print1("2, 3");forstep(n=3,1e4,6,if(isprime(t=n^2+2),print1(", "t))) \\ Charles R Greathouse IV, Jul 19 2011

For n>1, a(n) = 72*A000217(A056900(n-2))+11

a(n) = A067201(n)^2 + 2. - M. F. Hasler, Apr 05 2009

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

A106571 Indices n of perfect squares n^2 which are not the difference of two primes.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 25, 27, 29, 31, 35, 37, 41, 43, 47, 49, 51, 53, 55, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 149, 151, 153, 155

Offset: 1

Alexandre Wajnberg, May 09 2005

Also, n such that 1+n^2 is a nontotient (A005277). - T. D. Noe, Sep 13 2007

			a(3)=11 because the third square which is not the difference of two primes (121=11^2) is the 11th one in the succession of the perfect squares (thus index 11).

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A106544-A106548, A106562-A106564, A106573-A106575, A106577.

Cf. A067201 (n such that n^2 + 2 is prime).

a(n) = sqrt(A106564(n)).

Extended by Ray Chandler, May 12 2005

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

A114273 Numbers k such that k^2 + 10 is prime.

Original entry on oeis.org

1, 3, 7, 11, 13, 27, 31, 39, 49, 53, 57, 59, 71, 77, 81, 83, 91, 97, 99, 101, 123, 127, 129, 141, 151, 157, 161, 169, 171, 179, 181, 189, 207, 209, 211, 223, 227, 237, 239, 249, 253, 291, 311, 319, 333, 343, 363, 367, 379, 393, 403, 413, 423, 427, 437, 447, 449

Offset: 1

Zak Seidov, Nov 19 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1500

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), this sequence (i=10), A114274 (i=11), A114275 (i=12).

lst={}; Do[If[PrimeQ[n^2+10], AppendTo[lst, n]], {n, 600}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 26 2008 *)

is(n)=isprime(n^2+10) \\ 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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A056899 Primes of the form k^2 + 2.

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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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A106571 Indices n of perfect squares n^2 which are not the difference of two primes.

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

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