A088361 -id:A088361 - 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

A246397 Numbers n such that Phi(12, n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 3, 4, 5, 9, 10, 12, 13, 17, 25, 27, 30, 31, 36, 38, 39, 43, 48, 52, 55, 56, 61, 62, 65, 83, 92, 94, 99, 100, 104, 105, 109, 114, 118, 126, 131, 166, 168, 169, 172, 183, 185, 190, 194, 196, 198, 209, 224, 225, 229, 231, 239, 244, 257, 260, 261, 263, 269, 270, 272, 278, 291, 296, 299, 300, 302, 308, 311

Offset: 1

Eric Chen, Nov 13 2014

nonn

Numbers n such that n^4-n^2+1 is prime, or numbers n such that A060886(n) is prime.

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), this sequence (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).

Cf. A060886, A125258, A085398, A124990.

A246397:=n->`if`(isprime(n^4-n^2+1),n,NULL): seq(A246397(n),n=1..300); # Wesley Ivan Hurt, Nov 14 2014

Select[Range[350], PrimeQ[Cyclotomic[12, #]] &] (* Vincenzo Librandi, Jan 17 2015 *)

for(n=1,10^3,if(isprime(polcyclo(12,n)),print1(n,", "))); \\ Joerg Arndt, Nov 13 2014

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

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.

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

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

A277967 Number of even numbers b with 0 < b < 2^n such that b^(2^n) + 1 is prime.

Original entry on oeis.org

0, 1, 2, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 2, 3, 4, 1

Offset: 1

Jeppe Stig Nielsen, Nov 06 2016

The choice whether to take b < 2^n or b <= 2^n matters only for n=1 and n=2 unless there are more primes like 2^2+1 and 4^4+1 (see A121270).
Perfect squares b are allowed.
a(20) was determined after a lengthy computation by distributed project PrimeGrid, cf. A321323. - Jeppe Stig Nielsen, Jan 02 2019

			For n=18, we get b^262144 + 1 is prime for b=24518, 40734, 145310, 361658, 525094, ...; the first 3 of these b values are strictly below 262144, hence a(18)=3.
The corresponding primes are 2^4+1; 2^8+1, 4^8+1; 2^16+1; 30^32+1; 120^128+1; 46^512+1; etc.

Y. Gallot, Status of the smallest base values yielding Generalized Fermat primes.
PrimeGrid, GFN Prime Search Status and History.

Cf. A056993, A121270, A259835, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323, etc.

Table[Count[Range[2, 2^n - 1, 2], b_ /; PrimeQ[b^(2^n) + 1]], {n, 9}] (* Michael De Vlieger, Nov 10 2016 *)

a(n)=sum(k=1,2^(n-1)-1,ispseudoprime((2*k)^2^n+1)) \\ slow, only probabilistic primality test

a(20) from Jeppe Stig Nielsen, Jan 02 2019

cp's OEIS Frontend

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

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A246397 Numbers n such that Phi(12, n) is prime, where Phi is the cyclotomic polynomial.

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

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

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

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Mathematica

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

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

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Maple

Formula

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A272137 Primes of the form k^16 + 1.

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Magma

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A277967 Number of even numbers b with 0 < b < 2^n such that b^(2^n) + 1 is prime.

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