A005574 -id:A005574 - cp's OEIS frontend

A125261 Numbers k such that k^6 + 5 is prime.

0, 18, 24, 114, 204, 216, 222, 246, 276, 312, 318, 372, 384, 426, 438, 468, 498, 582, 618, 654, 822, 888, 948, 984, 1182, 1188, 1272, 1278, 1284, 1374, 1446, 1536, 1674, 1782, 1788, 1794, 1806, 1812, 1896, 2034, 2058, 2088, 2124, 2154, 2232, 2238, 2376

Offset: 1

Zak Seidov, Nov 26 2006

nonn

Carmine Suriano, Table of n, a(n) for n = 1..2254

Other sequences of the type "Numbers k such that k^j + j - 1 is prime": A000040 (j=1), A005574 (j=2), A067200 (j=3), A125259 (j=4), A125260 (j=5), this sequence(j=6), A125262 (j=7), A125263 (j=8), A125264 (j=10), A125265 (j=11)..

lst={};k=6;Do[If[PrimeQ[n^k+k-1], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)
Select[Range[0,2500],PrimeQ[#^6+5]&] (* Harvey P. Dale, Aug 29 2012 *)

is(n)=isprime(n^6+5) \\ Charles R Greathouse IV, Feb 17 2017

A125263 Numbers k such that k^8 + 7 is prime.

Original entry on oeis.org

0, 2, 4, 10, 66, 68, 86, 88, 134, 146, 200, 216, 250, 276, 306, 310, 410, 422, 472, 492, 506, 516, 538, 548, 550, 594, 638, 716, 746, 758, 862, 888, 942, 954, 964, 982, 992, 998, 1000, 1016, 1020, 1034, 1108, 1164, 1192, 1234, 1338, 1342, 1350, 1374, 1390

Offset: 1

Zak Seidov, Nov 26 2006

nonn

Carmine Suriano, Table of n, a(n) for n = 1..4683

Other sequences of the type "Numbers k such that k^j + j - 1 is prime": A000040 (j=1), A005574 (j=2), A067200 (j=3), A125259 (j=4), A125260 (j=5), A125261 (j=6), A125262 (j=7), this sequence (j=8), A125264 (j=10), A125265 (j=11)...

lst={};k=8;Do[If[PrimeQ[n^k+k-1], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)
Select[Range[0,1500],PrimeQ[#^8+7]&] (* Harvey P. Dale, Sep 04 2024 *)

is(n)=isprime(n^8+7) \\ Charles R Greathouse IV, Feb 17 2017

A125264 Numbers k such that k^10 + 9 is prime.

Original entry on oeis.org

2, 8, 238, 310, 338, 442, 542, 688, 698, 872, 920, 1198, 1330, 1382, 1538, 1558, 1678, 1702, 1712, 1768, 1882, 2032, 2080, 2102, 2260, 2312, 2408, 2440, 2540, 2642, 3112, 3170, 3188, 3268, 3320, 3580, 3740, 3742, 3770, 3980, 4028, 4048, 4148, 4292, 4472

Offset: 1

Zak Seidov, Nov 26 2006

nonn

Carmine Suriano, Table of n, a(n) for n = 1..1524

Other sequences of the type "Numbers k such that k^j + j - 1 is prime": A000040 (j=1), A005574 (j=2), A067200 (j=3), A125259 (j=4), A125260 (j=5), A125261 (j=6), A125262 (j=7), A125263 (j=8), this sequence (j=10), A125265 (j=11).

lst={};k=10;Do[If[PrimeQ[n^k+k-1], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)
Select[Range[5000],PrimeQ[#^10+9]&] (* Harvey P. Dale, Nov 17 2014 *)

is(n)=isprime(n^10+9) \\ Charles R Greathouse IV, Feb 17 2017

A135590 Numbers k such that k^2 + 1 is a Sarrus number (pseudoprime to base 2).

Original entry on oeis.org

216, 948, 1560, 4872, 8208, 9828, 18200, 29640, 37024, 65536, 89550, 283800, 535920, 592956, 649800, 825930, 1042320, 1382400, 1536220, 3688230, 4215120, 4321800, 5103210, 19078930, 21415680, 24471720, 214067490, 435457620, 535019100

Offset: 1

Jason Earls, Feb 25 2008

nonn

Note that A000215(5) corresponds to a(10), and A000215(6) corresponds to a(33), and in general when A000215(n) is composite, this sequence has corresponding entry. - Jeppe Stig Nielsen, Mar 26 2016

Amiram Eldar, Table of n, a(n) for n = 1..40 (terms 1..33 from Jeppe Stig Nielsen)

Cf. A000215, A001567, A002522, A005574.

fQ[n_] := ( !PrimeQ[n^2 + 1] && PowerMod[2, n^2, n^2 + 1] == 1); lst = {}; Do[ If[ fQ@ n, AppendTo[lst, n]], {n, 2, 440000000, 2}]; lst (* Robert G. Wilson v, Apr 18 2008 *)

is(n) = {Mod(2, n)^(n-1)==1 && !ispseudoprime(n) && n > 1};
for(n=1, 1e10, if(is(n^2+1), print1(n, ", "))); \\ Altug Alkan, Mar 26 2016

More terms from Robert G. Wilson v, Apr 18 2008

A083847 a(n) = number of primes of the form x^2 + 1 <= 2^n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 14, 18, 24, 33, 42, 54, 70, 91, 114, 158, 212, 293, 393, 539, 713, 957, 1301, 1792, 2459, 3378, 4615, 6233, 8418, 11540, 15867, 21729, 29843, 41169, 56534, 77697, 106787, 147067, 203025, 280340, 387308, 535153, 739671, 1023655, 1416635, 1960813, 2716922, 3764693, 5218926, 7238715

Offset: 1

Harry J. Smith, May 05 2003

nonn

It is conjectured that the number of primes of the form x^2+1 is infinite and thus this sequence does not become a constant, but this has never been proved.

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.
Eric Weisstein's World of Mathematics, Landau's Problems.

Cf. A005574, A002496, A083844 - A083849.

a(n) = my(nb = 0); forprime(p=2, 2^n, if (issquare(p-1), nb++);); nb  \\ Michel Marcus, Jun 14 2013

More terms from Alexander D. Healy, Feb 06 2005

A121250 Numbers n such that n^2 + 14 is prime.

Original entry on oeis.org

3, 15, 27, 33, 45, 75, 87, 93, 165, 183, 195, 207, 243, 285, 297, 303, 345, 363, 375, 405, 435, 453, 495, 513, 537, 573, 585, 615, 627, 633, 657, 663, 717, 813, 843, 975, 1053, 1065, 1083, 1095, 1125, 1137, 1167, 1203, 1287, 1317, 1335, 1353, 1413, 1437, 1455

Offset: 1

Parthasarathy Nambi, Sep 06 2006

nonn

			If n=183 then n^2 + 14 = 33503 (prime).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A005574, A067201, A049422, A007591, A078402, A114269, A114271, A114272, A114273, A114274, A114275, A113536.

[n: n in [0..6000] | IsPrime(n^2+14)] // Vincenzo Librandi, Sep 03 2010

Select[Range[1,1500,2],PrimeQ[#^2+14]&] (* Harvey P. Dale, Aug 20 2011 *)

is(n)=isprime(n^2+14) \\ Charles R Greathouse IV, Feb 20 2017

A123599 Smallest generalized Fermat prime of the form a^(2^n) + 1, where base a>1 is an integer; or -1 if no such prime exists.

Original entry on oeis.org

3, 5, 17, 257, 65537, 185302018885184100000000000000000000000000000001

Offset: 0

Alexander Adamchuk, Nov 14 2006

nonn

First 5 terms {3, 5, 17, 257, 65537} = A019434 are the Fermat primes of the form 2^(2^n) + 1. Note that for all currently known a(n) up to n = 17 last digit is 7 or 1 (except a(0) = 3 and a(1) = 5). Corresponding least bases a>1 such that a^(2^n) + 1 is prime are listed in A056993.

The last-digit behavior clearly continues since, for any a, we have that a^(2^2) will be either 0 or 1 modulo 5. So for n >= 2, a(n) is 1 or 2 modulo 5, and odd. - Jeppe Stig Nielsen, Nov 16 2020

Charles R Greathouse IV, Table of n, a(n) for n = 0..9
Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Cf. A000215, A019434, A056993.

Cf. A006093, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002.

Do[f=Min[Select[ Table[ i^(2^n) + 1, {i, 2, 500} ],PrimeQ]];Print[{n,f}],{n,0,9}]

A180252 Numbers where all prime divisors are of the form k^2+1.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 16, 17, 20, 25, 32, 34, 37, 40, 50, 64, 68, 74, 80, 85, 100, 101, 125, 128, 136, 148, 160, 170, 185, 197, 200, 202, 250, 256, 257, 272, 289, 296, 320, 340, 370, 394, 400, 401, 404, 425, 500, 505, 512, 514

Offset: 1

Michel Lagneau, Jan 20 2011

nonn

			a(17) = 74 because 74 = 2*37 = (1^2+1)*(6^2+1).

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A002496, A002522, A005574.

with(numtheory):T:=array(1..50):U:=array(1..1000):k:=1:for m from 1 to 300
  do:x:=m^2+1:if type(x,prime)=true then T[k]:=x:k:=k+1:else fi:od:for x from
  2 to 2000 do: B:=factorset(x):yy:=nops(B):A:=convert(T, set):if A intersect
  B = B then printf(`%d, `, x):else fi:od:

Select[Range@520, And @@ IntegerQ /@ Sqrt[FactorInteger[#][[All, 1]] - 1] &] (* Ivan Neretin, Aug 31 2016 *)

Sum_{n>=1} 1/a(n) = Product_{p in A002496} p/(p-1) = Product_{k in A005574} (1 + 1/k^2) = 2.809865... - Amiram Eldar, Sep 27 2020

A250177 Numbers n such that Phi_21(n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

3, 6, 7, 12, 22, 27, 28, 35, 41, 59, 63, 69, 112, 127, 132, 133, 136, 140, 164, 166, 202, 215, 218, 276, 288, 307, 323, 334, 343, 377, 383, 433, 474, 479, 516, 519, 521, 532, 538, 549, 575, 586, 622, 647, 675, 680, 692, 733, 790, 815, 822, 902, 909, 911, 915, 952, 966, 1025, 1034, 1048, 1093

Offset: 1

Eric Chen, Dec 24 2014

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A008864 (1), A006093 (2), A002384 (3), A005574 (4), A049409 (5), A055494 (6), A100330 (7), A000068 (8), A153439 (9), A250392 (10), A162862 (11), A246397 (12), A217070 (13), A250174 (14), A250175 (15), A006314 (16), A217071 (17), A164989 (18), A217072 (19), A250176 (20), this sequence (21), A250178 (22), A217073 (23), A250179 (24), A250180 (25), A250181 (26), A153440 (27), A250182 (28), A217074 (29), A250183 (30), A217075 (31), A006313 (32), A250184 (33), A250185 (34), A250186 (35), A097475 (36), A217076 (37), A250187 (38), A250188 (39), A250189 (40), A217077 (41), A250190 (42), A217078 (43), A250191 (44), A250192 (45), A250193 (46), A217079 (47), A250194 (48), A250195 (49), A250196 (50), A217080 (53), A217081 (59), A217082 (61), A006315 (64), A217083 (67), A217084 (71), A217085 (73), A217086 (79), A153441 (81), A217087 (83), A217088 (89), A217089 (97), A006316 (128), A153442 (243), A056994 (256), A056995 (512), A057465 (1024), A057002 (2048), A088361 (4096), A088362 (8192), A226528 (16384), A226529 (32768), A226530 (65536), A251597 (131072), A244150 (524287), A243959 (1048576).

Cf. A085398 (Least k>1 such that Phi_n(k) is prime).

a250177[n_] := Select[Range[n], PrimeQ@Cyclotomic[21, #] &]; a250177[1100] (* Michael De Vlieger, Dec 25 2014 *)

{is(n)=isprime(polcyclo(21,n))};
for(n=1,100, if(is(n)==1, print1(n, ", "), 0)) \\ G. C. Greubel, Apr 14 2018

A259645 Numbers m such that m^2 + 1, 3*m - 1 and m^2 + m + 41 are all prime.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 16, 20, 24, 36, 66, 90, 94, 116, 120, 134, 150, 156, 160, 206, 240, 280, 340, 350, 384, 396, 430, 436, 470, 536, 634, 690, 700, 714, 780, 826, 864, 930, 946, 960, 1004, 1124, 1150, 1176, 1294, 1316, 1376, 1410, 1430, 1494, 1644, 1674

Offset: 1

Reinhard Zumkeller, Jul 03 2015

nonn

This sequence is infinite if the generalized Dickson's conjecture holds.

			.            | (i, j, k) such that |        corresponding
.            | a(n) = A005574(i)   |        prime triples
.     |      |      = A087370(j)   |        let m = a(n):
.   n | a(n) |      = A056561(k)   |  (m^2+1, 3*m-1, m^2+m+41)
.  ---+------+---------------------+--------------------------
.   1 |    1 |     (1,  1,  2)     |        (2,   2,   43)
.   2 |    2 |     (2,  2,  3)     |        (5,   5,   47)
.   3 |    4 |     (3,  3,  5)     |       (17,  11,   61)
.   4 |    6 |     (4,  4,  7)     |       (37,  17,   83)
.   5 |   10 |     (5,  6, 11)     |      (101,  29,  151)
.   6 |   14 |     (6,  7, 13)     |      (197,  41,  251)
.   7 |   16 |     (7,  8, 15)     |      (257,  47,  313)
.   8 |   20 |     (8, 10, 21)     |      (401,  59,  461)
.   9 |   24 |     (9, 11, 25)     |      (597,  71,  641)
.  10 |   36 |    (11, 15, 37)     |     (1297, 107, 1373)
.  11 |   66 |    (15, 24, 61)     |     (4357, 197, 4463)
.  12 |   90 |    (18, 31, 79)     |     (8101, 269, 8231)  .

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Wikipedia, Bunyakovsky conjecture
Wikipedia, Dickson's conjecture

Intersection of A005574, A087370 and A056561.

import Data.List.Ordered (isect)
a259645 n = a259645_list !! (n-1)
a259645_list = a005574_list `isect` a087370_list `isect` a056561_list

Select[Range[100], AllTrue[{#^2 + 1, 3 # - 1, #^2 + # + 41}, PrimeQ] &] (* Robert Price, Apr 19 2025 *)

cp's OEIS Frontend

A125261 Numbers k such that k^6 + 5 is prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A125263 Numbers k such that k^8 + 7 is prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A125264 Numbers k such that k^10 + 9 is prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A135590 Numbers k such that k^2 + 1 is a Sarrus number (pseudoprime to base 2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A083847 a(n) = number of primes of the form x^2 + 1 <= 2^n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

PARI

Extensions

A121250 Numbers n such that n^2 + 14 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A123599 Smallest generalized Fermat prime of the form a^(2^n) + 1, where base a>1 is an integer; or -1 if no such prime exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A180252 Numbers where all prime divisors are of the form k^2+1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica