A005574 -id:A005574 - cp's OEIS frontend

A321323 Numbers k such that k^(2^20) + 1 is prime (a generalized Fermat prime).

1, 919444, 1059094, 1951734, 1963736, 3843236

Offset: 1

Jeppe Stig Nielsen, Nov 04 2018

Chris K. Caldwell, The Prime Database: 919444^1048576 + 1
Chris K. Caldwell, The Prime Database: 1059094^1048576 + 1
Chris K. Caldwell, The Prime Database: ?^1048576 + 1

Cf. A056993, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959.

a(4) from Jeppe Stig Nielsen, Aug 31 2022
a(5) from Jeppe Stig Nielsen, Oct 21 2022
a(6) from Jeppe Stig Nielsen, Jan 11 2025

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

A134406 Composite numbers of the form k^2 + 1.

Original entry on oeis.org

10, 26, 50, 65, 82, 122, 145, 170, 226, 290, 325, 362, 442, 485, 530, 626, 730, 785, 842, 901, 962, 1025, 1090, 1157, 1226, 1370, 1445, 1522, 1682, 1765, 1850, 1937, 2026, 2117, 2210, 2305, 2402, 2501, 2602, 2705, 2810, 3026, 3250, 3365, 3482, 3601, 3722, 3845

Offset: 1

Jani Melik, Jan 18 2008

nonn

Square roots of these numbers are quadratic irrationals and corresponding chain fraction representations are periodic: sqrt(10) = [3;{2,3}], sqrt(26) = [5;{2,5}], sqrt(50) = [7;{2,7}], ..., where {} is denoted a period (we write {6} == {2,3}).

			10 is a term because 10 = 3^2 + 1 is composite,
26 is a term because 26 = 5^2 + 1 is composite,
50 is a term because 50 = 7^2 + 1 is composite.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002496, A005574, A134407.

Supersequence of A144255.

ts_fn1:=proc(n) local i,tren,ans; ans:=[ ]: for i from 1 to n do tren := i^(2)+1: if (isprime(tren) = false) then ans:=[ op(ans), tren ]: fi od: RETURN(ans) end: ts_fn1(200);

Select[Range[70]^2+1,!PrimeQ[#]&] (* Harvey P. Dale, Aug 12 2012 *)

for(n=3,99, if(!isprime(t=n^2+1), print1(t", "))) \\ Charles R Greathouse IV, Aug 29 2016

from sympy import isprime
from itertools import count, takewhile
def aupto(limit):
    form = takewhile(lambda x: x <= limit, (k**2+1 for k in count(1)))
    return [number for number in form if not isprime(number)]
print(aupto(3845)) # Michael S. Branicky, Oct 26 2021

a(n) = 1 + A134407(n)^2. - R. J. Mathar, Oct 13 2019

A083844 Number of primes of the form x^2 + 1 < 10^n.

Original entry on oeis.org

2, 4, 10, 19, 51, 112, 316, 841, 2378, 6656, 18822, 54110, 156081, 456362, 1339875, 3954181, 11726896, 34900213, 104248948, 312357934, 938457801, 2826683630, 8533327397, 25814570672, 78239402726, 237542444180, 722354138859, 2199894223892

Offset: 1

Harry J. Smith, May 05 2003

nonn

It is conjectured that there are infinitely many primes of the form x^2 + 1 (and thus this sequence never becomes constant), but this has not been proved.

These primes can be found quickly using a sieve based on the fact that numbers of this form have at most one primitive prime factor (A005529). The sum of the reciprocals of these primes is 0.81459657... - T. D. Noe, Oct 14 2003

			a(3) = 10 because the only primes or the form x^2 + 1 < 10^3 are the ten primes: 2, 5, 17, 37, 101, 197, 257, 401, 577, 677.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.
P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 190.

C. K. Caldwell, An Amazing Prime Heuristic A pdf file.
Jon Grantham, Parallel Computation of Primes of the form x^2+1
Jon Grantham and Hester Graves, Primes of the Form m^2+1 and Goldbach's 'Other Other' Conjecture, arXiv:2502.03513 [math.NT], 2025.
Apoloniusz Tyszka, Sławomir Kurpaska, Open problems that concern computable sets X, subset of N, and cannot be formally stated as they refer to current knowledge about X, (2020).
Eric Weisstein's World of Mathematics, Landau's Problems.
Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Cf. A005574, A002496, A083845, A083846, A083847, A083848, A083849, A214452, A214454, A214455.

Cf. A005529 (primitive prime factors of the sequence k^2+1).

c = 1; k = 2; (* except for the initial prime 2, all X's must be odd. *) Do[ While[ k^2 + 1 < 10^n, If[ PrimeQ[k^2 + 1], c++ ]; k += 2]; Print[c], {n, 1, 20}]

Edited by Robert G. Wilson v, May 08 2003
More terms from T. D. Noe, Oct 14 2003
a(17)-a(22) from Robert Gerbicz, Apr 15 2009
a(23)-a(25) from Marek Wolf and Robert Gerbicz (code from Robert, computation done by Marek) Robert Gerbicz, Mar 13 2010
a(26)-a(28) from Jon Grantham, Jan 18 2017
a(28) corrected by Jon Grantham, Jan 30 2018

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

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

A321323 Numbers k such that k^(2^20) + 1 is prime (a generalized Fermat prime).

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

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A134406 Composite numbers of the form k^2 + 1.

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A083844 Number of primes of the form x^2 + 1 < 10^n.

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

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

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