A006313 -id:A006313 - cp's OEIS frontend

A070020 At these values of k the first, 2nd and 3rd cyclotomic polynomials all give prime numbers.

6, 12, 138, 150, 192, 348, 642, 1020, 1092, 1230, 1620, 1788, 1932, 2112, 2142, 2238, 2658, 2688, 2730, 3330, 3540, 3918, 4002, 4158, 5010, 5640, 6090, 6450, 6552, 6702, 7950, 8088, 9000, 9042, 9240, 9462, 9768, 10008, 10092, 10272, 10302, 10332

Offset: 1

Labos Elemer, May 07 2002

Numbers k such that k-1, k+1 and k^2+k+1 are all primes.

			For k = 6: 5, 7 and 43 are prime values of the first 3 cyclotomic polynomials.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Michael De Vlieger)

Cf. A070155, A070156, A070157, A000068, A006313, A006314, A006315, A006316, A056993, A056994, A056995, A005574, A057465, A057002, A070025, A070042.

psQ[n_]:=And@@PrimeQ[{n-1,n+1,n^2+n+1}]; Select[Range[11000],psQ] (* Harvey P. Dale, Nov 05 2011 *)
Select[Range[10500], AllTrue[Cyclotomic[Range@ 3, #], PrimeQ] &] (* Michael De Vlieger, Dec 08 2018 *)

is(k) = isprime(k-1) && isprime(k+1) && isprime(k^2+k+1); \\ Amiram Eldar, Sep 24 2024

A070042 At these values of k the 1st, 2nd, 3rd, 4th and 5th cyclotomic polynomials all give prime numbers.

Original entry on oeis.org

1068630, 1441590, 1867950, 3429300, 4084230, 5651730, 6322890, 6770610, 7158630, 7804830, 9437760, 9624270, 13625850, 23194860, 25848840, 26588520, 28714950, 29451840, 32984430, 33650580, 36500910, 38177130, 42856590, 49531020, 50016540, 50222070, 52083330, 54637590

Offset: 1

Labos Elemer, May 07 2002

Numbers k such that C1(k) = k-1, C2(k) = k+1, C3(k) = k^2+k+1, C4(k) = k^2+1 and C5(k) = k^4+k^3+k^2+k+1 are all primes.

			For k = 1068630: the 1st, 2nd, 3rd, 4th and 5th cyclotomic polynomials give a quintet of primes: {1068629, 1068631, 1141971145531, 1141970076901, 1304096876879617162402531}.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A070155, A070156, A070157, A000068, A006313, A006314, A006315, A006316, A056993, A056994, A056995, A005574, A057465, A057002, A070020, A070025.

is(k) = isprime(k-1) && isprime(k+1) && isprime(k^2+1) && isprime(k^2+k+1) && isprime(k^4+k^3+k^2+k+1) ; \\ Amiram Eldar, Sep 24 2024

More terms from Don Reble, May 11 2002

a(24)-a(28) from Amiram Eldar, Sep 24 2024

A250175 Numbers n such that Phi_15(n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 3, 11, 17, 23, 43, 46, 52, 53, 61, 62, 78, 84, 88, 89, 92, 99, 108, 123, 124, 141, 146, 154, 156, 158, 163, 170, 171, 182, 187, 202, 217, 219, 221, 229, 233, 238, 248, 249, 253, 264, 274, 275, 278, 283, 285, 287, 291, 296, 302, 309, 314, 315, 322, 325, 342, 346, 353, 356, 366, 368, 372, 377, 380, 384, 394, 404, 406, 411, 420, 425

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), A246392 (10), A162862(11), A246397 (12), A217070 (13), A006314 (16), A217071 (17), A164989(18), A217072 (19), A217073 (23), A153440 (27), A217074 (29), A217075(31), A006313 (32), A097475 (36), A217076 (37), A217077 (41), A217078(43), A217079 (47), 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).

Select[Range[600], PrimeQ[Cyclotomic[15, #]] &] (* Vincenzo Librandi, Jan 16 2015 *)

isok(n) = isprime(polcyclo(15, n)); \\ Michel Marcus, Jan 16 2015

More terms from Vincenzo Librandi, Jan 16 2015

A250176 Numbers n such that Phi_20(n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

4, 9, 11, 16, 19, 26, 34, 45, 54, 70, 86, 91, 96, 101, 105, 109, 110, 119, 120, 126, 129, 139, 141, 149, 171, 181, 190, 195, 215, 229, 260, 276, 299, 305, 309, 311, 314, 319, 334, 339, 369, 375, 414, 420, 425, 444, 470, 479, 485, 506, 519, 534, 540, 550

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), A246392 (10), A162862(11), A246397 (12), A217070 (13), A006314 (16), A217071 (17), A164989(18), A217072 (19), A217073 (23), A153440 (27), A217074 (29), A217075(31), A006313 (32), A097475 (36), A217076 (37), A217077 (41), A217078(43), A217079 (47), 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).

Select[Range[600], PrimeQ[Cyclotomic[20, #]] &] (* Vincenzo Librandi, Jan 16 2015 *)

isok(n) = isprime(polcyclo(20, n)); \\ Michel Marcus, Sep 29 2015

More terms from Vincenzo Librandi, Jan 16 2015

A087738 Square array: T(n,k) gives n-th number a such that a^(2^k)+1 is prime (a generalized Fermat).

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 6, 4, 2, 1, 10, 6, 4, 2, 1, 12, 10, 6, 4, 2, 1, 16, 14, 16, 118, 44, 30, 1, 18, 16, 20, 132, 74, 54, 102, 1, 22, 20, 24, 140, 76, 96, 162, 120, 1, 28, 24, 28, 152, 94, 112, 274, 190, 278, 1, 30, 26, 34, 208, 156, 114, 300, 234, 614, 46, 1, 36, 36, 46, 240, 158

Offset: 0

Jeppe Stig Nielsen, Oct 01 2003

			{1}; {2,1}; {4,2,1}; ...
See the well-formed array on Gallot's page.

Harvey Dubner, J. Recr. Math., 18, 1986.

Yves Gallot, Generalized Fermat Prime Search.

Cf. A056993, A006093, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362.

A186689 Numbers n such that n^4 + 1 is a semiprime.

Original entry on oeis.org

3, 5, 7, 8, 10, 11, 12, 13, 14, 17, 18, 21, 22, 23, 26, 29, 30, 32, 35, 36, 38, 39, 40, 42, 50, 52, 57, 58, 61, 62, 65, 68, 71, 72, 73, 78, 81, 84, 86, 92, 94, 98, 100, 102, 103, 105, 108, 112, 113, 114, 115, 116, 119, 120, 122, 124, 128, 129, 130, 138, 146, 148, 152, 153, 158

Offset: 1

Michel Lagneau, Feb 25 2011

nonn

Corresponding semiprimes n^4+1 are in A186688.

			3 is in the sequence because 3^4 + 1 = 82 = 2*41 is semiprime.

Robert Price, Table of n, a(n) for n = 1..1500

Cf. A001358, A006313, A085722, A103854, A104238, A104335, A105041, A105066, A105078, A105122, A105142, A105237, A104479, A104494, A104657, A105282, A186688.

SemiPrimeQ[ n_] := (n > 1) && (2 == Plus @@ (Transpose[FactorInteger[n]][[2]]));
  Select[Range[300], SemiPrimeQ[#^4 + 1] &]
Select[Range[200],PrimeOmega[#^4+1]==2&] (* Harvey P. Dale, Jan 27 2013 *)

A217993 Smallest k such that k^(2^n) + 1 and (k+2)^(2^n) + 1 are both prime.

Original entry on oeis.org

2, 2, 2, 2, 74, 112, 2162, 63738, 13220, 54808, 3656570, 6992032, 125440, 103859114, 56414914, 87888966

Offset: 0

Michel Lagneau, Oct 17 2012

a(15)=87888966 but a(14) is unknown. - Jeppe Stig Nielsen, Mar 17 2018

The prime pair related to a(14) was found four days ago, and today double checking has proved that they are indeed the first occurrence for n=14. - Jeppe Stig Nielsen, May 02 2018

			a(0) = 2 because 2^1+1 = 3 and 4^1+1 = 5 are prime;
a(1) = 2 because 2^2+1 = 5  and 4^2+1 = 17 are prime;
a(2) = 2 because 2^4+1 = 17  and 4^4+1 = 257 are prime;
a(3) = 2 because  2^8+1 = 257 and 4^8+1 = 65537 are prime.

Cf. A006313, A006314, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A118539.

for n from 0  to 5 do:ii:=0:for k from 2 by 2 to 10000 while(ii=0) do:if type(k^(2^n)+1,prime)=true and type((k+2)^(2^n)+1,prime)=true then ii:=1: printf ( "%d %d \n",n,k):else fi:od:od:

a(n) = A118539(n)-1. - Jeppe Stig Nielsen, Feb 27 2016

a(13) from Jeppe Stig Nielsen, Mar 17 2018

a(14) and a(15) from Jeppe Stig Nielsen, May 02 2018

A242554 Least number k such that n^16 + k^16 is prime.

Original entry on oeis.org

1, 1, 4, 3, 6, 5, 6, 7, 22, 13, 16, 5, 8, 5, 14, 11, 10, 7, 16, 31, 8, 9, 10, 11, 38, 29, 10, 9, 22, 61, 20, 5, 4, 3, 16, 11, 6, 25, 28, 7, 6, 17, 16, 1, 46, 9, 58, 61, 22, 41, 92, 3, 14, 19, 14, 23, 56, 37, 20, 109, 6, 121, 10, 39, 4, 67, 34, 11, 26, 9, 30, 11, 12, 1

Offset: 1

Derek Orr, May 17 2014

nonn

If a(n) = 1, then n is in A006313.

			4^16+1^16 = 4294967297 is not prime. 4^16+2^16 = 4295032832 is not prime. 4^16+3^16 = 4338014017 is prime. Thus, a(4) = 3.

Cf. A069003, A006313.

a(n)=for(k=1,oo,if(ispseudoprime(n^16+k^16),return(k)));

import sympy
from sympy import isprime
def a(n):
  for k in range(10**4):
    if isprime(n**16+k**16):
      return k
n = 1
while n < 100:
  print(a(n))
  n += 1

A272137 Primes of the form k^16 + 1.

Original entry on oeis.org

2, 65537, 197352587024076973231046657, 808551180810136214718004658177, 1238846438084943599707227160577, 37157429083410091685945089785857, 123025056645280288014028950372089857, 150838912030874130174020868290707457

Offset: 1

Jaroslav Krizek, May 08 2016

nonn

Corresponding values of k are in A006313.

Jaroslav Krizek, Table of n, a(n) for n = 1..100

Cf. Sequences of numbers n such that n^(2^k)+1 is a prime p for k = 1-13: A005574 (k=1), A000068 (k=2), A006314 (k=3), A006313 (k=4), A006315 (k=5), A006316 (k=6), A056994 (k=7), A056995 (k=8), A057465 (k=9), A057002 (k=10), A088361 (k=11), A088362 (k=12), A226528 (k=13).

Corresponding sequences of primes p of the form n^(2^k)+1 for k = 1-4: A002496 (k=1), A037896 (k=2), A258805 (k=3), A272137 (k=4).

[n^16 + 1: n in [1..700] | IsPrime(n^16 + 1)];

A272137:=n->`if`(isprime(n^16+1), n^16+1, NULL): seq(A272137(n), n=1..200); # Wesley Ivan Hurt, May 11 2016

A105934 Positive integers n such that n^22 + 1 is semiprime (A001358).

Original entry on oeis.org

116, 176, 184, 300, 444, 470, 584, 690, 696, 950

Offset: 1

Jonathan Vos Post, Apr 26 2005

We have the polynomial factorization: n^22 + 1 = (n^2 + 1) * (n^20 - n^18 + n^16 - n^14 + n^12 - n^10 + n^8 - n^6 + n^4 - n^2 + 1). Hence after the initial n=1 prime, the binomial can never be prime. It can be semiprime iff n^2+1 is prime and (n^20 - n^18 + n^16 - n^14 + n^12 - n^10 + n^8 - n^6 + n^4 - n^2 + 1) is prime.

			116^22 + 1 = 2618639792014920380336685706161496723088736257 = 13457 * 194593133091693570657404005808240820620401,
300^22 + 1 = 3138105960900000000000000000000000000000000000000000001 = 90001 * 34867456593815624270841435095165609271008099910001,
950^22 + 1 = 323533544973709366507562922501564025878906250000000000000000000001 = 902501 * 358485525194663902319845543109164450653136395416736380347501.

Cf. A000040, A001358, A006313, A103854, A104238, A104335, A105041, A105066, A105078, A105122, A105142, A105237, A104479, A104494, A104657, A105282.

IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [2..1000] | IsSemiprime(n^22+1)]; // Vincenzo Librandi, Dec 21 2010

Select[Range[1000], PrimeOmega[#^22 + 1]==2&] (* Vincenzo Librandi, May 24 2014 *)

a(n)^22 + 1 is in A001358. a(n)^2+1 is in A000040 and (a(n)^20 - a(n)^18 + a(n)^16 - a(n)^14 + a(n)^12 - a(n)^10 + a(n)^8 - a(n)^6 + a(n)^4 - a(n)^2 + 1) is in A000040.

cp's OEIS Frontend

A070020 At these values of k the first, 2nd and 3rd cyclotomic polynomials all give prime numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A070042 At these values of k the 1st, 2nd, 3rd, 4th and 5th cyclotomic polynomials all give prime numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A250175 Numbers n such that Phi_15(n) is prime, where Phi is the cyclotomic polynomial.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A250176 Numbers n such that Phi_20(n) is prime, where Phi is the cyclotomic polynomial.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A087738 Square array: T(n,k) gives n-th number a such that a^(2^k)+1 is prime (a generalized Fermat).

Views

Author

Keywords

Examples

References

Links

Crossrefs

A186689 Numbers n such that n^4 + 1 is a semiprime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A217993 Smallest k such that k^(2^n) + 1 and (k+2)^(2^n) + 1 are both prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

Extensions

A242554 Least number k such that n^16 + k^16 is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs