A006314 -id:A006314

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A321323

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

+0
21

1, 919444, 1059094, 1951734, 1963736

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..5.

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

CROSSREFS

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

KEYWORD

nonn,hard,more

AUTHOR

Jeppe Stig Nielsen, Nov 04 2018

EXTENSIONS

a(4) from Jeppe Stig Nielsen, Aug 31 2022

a(5) from Jeppe Stig Nielsen, Oct 21 2022

STATUS

approved

A277967

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

+0
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

LINKS

Table of n, a(n) for n=1..20.

Y. Gallot, Status of the smallest base values yielding Generalized Fermat primes.

PrimeGrid, GFN Prime Search Status and History.

EXAMPLE

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.

MATHEMATICA

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

PROG

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

CROSSREFS

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.

KEYWORD

nonn,hard,more

AUTHOR

Jeppe Stig Nielsen, Nov 06 2016

EXTENSIONS

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

STATUS

approved

A272137

Primes of the form n^16 + 1.

+0
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Corresponding values of n are in A006313.

LINKS

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

MAPLE

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

PROG

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

CROSSREFS

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).

KEYWORD

nonn

AUTHOR

Jaroslav Krizek, May 08 2016

STATUS

approved

A258805

Primes of the form n^8 + 1.

+0
2

2, 257, 65537, 37588592026706177, 92170395205042177, 147578905600000001, 284936905588473857, 3503536769037500417, 11007531417600000001, 11763130845074473217, 47330370277129322497, 50024641296100000001, 76872571987558646017, 416806419029812551937

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = A060890(A006314(n)). - Michel Marcus, Jun 11 2015

MATHEMATICA

Select[Range[500]^8 + 1, PrimeQ]

PROG

(Magma) [a: n in [1..500] | IsPrime(a) where a is n^8+1];

(PARI) is(n)=ispower(n-1, 8) && isprime(n) \\ Charles R Greathouse IV, Jun 11 2015

CROSSREFS

Subsequence of A002496, A037896.

Cf. A006314 (associated n), A060890.

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, Jun 11 2015

STATUS

approved

A253240

Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

+0
3

1, 1, -1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 3, 4, 7, 2, 1, 1, 4, 5, 13, 5, 5, 1, 1, 5, 6, 21, 10, 31, 1, 1, 1, 6, 7, 31, 17, 121, 3, 7, 1, 1, 7, 8, 43, 26, 341, 7, 127, 2, 1, 1, 8, 9, 57, 37, 781, 13, 1093, 17, 3, 1, 1, 9, 10, 73, 50, 1555, 21, 5461, 82, 73, 1, 1, 1, 10, 11, 91, 65, 2801, 31, 19531, 257, 757, 11, 11, 1, 1, 11, 12, 111, 82, 4681, 43, 55987, 626, 4161, 61, 2047, 1, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,9

COMMENTS

Outside of rows 0, 1, 2 and columns 0, 1, only terms of A206942 occur.

Conjecture: There are infinitely many primes in every row (except row 0) and every column (except column 0), the indices of the first prime in n-th row and n-th column are listed in A117544 and A117545. (See A206864 for all the primes apart from row 0, 1, 2 and column 0, 1.)

Another conjecture: Except row 0, 1, 2 and column 0, 1, the only perfect powers in this table are 121 (=Phi_5(3)) and 343 (=Phi_3(18)=Phi_6(19)).

LINKS

Eric Chen, Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)

Eric Weisstein's World of Mathematics, Cyclotomic polynomial

Wikipedia, Cyclotomic polynomial

Index entries for Cyclotomic polynomials, values at X

FORMULA

T(m, n) = Phi_m(n)

EXAMPLE

Read by antidiagonals:

m\n 0 1 2 3 4 5 6 7 8 9 10 11 12

------------------------------------------------------

0 1 1 1 1 1 1 1 1 1 1 1 1 1

1 -1 0 1 2 3 4 5 6 7 8 9 10 11

2 1 2 3 4 5 6 7 8 9 10 11 12 13

3 1 3 7 13 21 31 43 57 73 91 111 133 157

4 1 2 5 10 17 26 37 50 65 82 101 122 145

5 1 5 31 121 341 781 ... ... ... ... ... ... ...

6 1 1 3 7 13 21 31 43 57 73 91 111 133

etc.

The cyclotomic polynomials are:

n n-th cyclotomic polynomial

0 1

1 x-1

2 x+1

3 x^2+x+1

4 x^2+1

5 x^4+x^3+x^2+x+1

6 x^2-x+1

...

MATHEMATICA

Table[Cyclotomic[m, k-m], {k, 0, 49}, {m, 0, k}]

PROG

(PARI) t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2)

t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)

T(m, n) = if(m==0, 1, polcyclo(m, n))

a(n) = T(t1(n), t2(n))

CROSSREFS

Rows 0-16 are A000012, A023443, A000027, A002061, A002522, A053699, A002061, A053716, A002523, A060883, A060884, A060885, A060886, A060887, A060888, A060889, A060890.

Columns 0-13 are A158388, A020500, A019320, A019321, A019322, A019323, A019324, A019325, A019326, A019327, A019328, A019329, A019330, A019331.

Main diagonal is A070518.

Indices of primes in n-th row for n = 1-20 are A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A250174, A250175, A006314, A217071, A164989, A217072, A250176.

Indices of primes in n-th column for n = 1-10 are A246655, A072226, A138933, A138934, A138935, A138936, A138937, A138938, A138939, A138940.

Indices of primes in main diagonal is A070519.

Cf. A117544 (indices of first prime in n-th row), A085398 (indices of first prime in n-th row apart from column 1), A117545 (indices of first prime in n-th column).

Cf. A206942 (all terms (sorted) for rows>2 and columns>1).

Cf. A206864 (all primes (sorted) for rows>2 and columns>1).

KEYWORD

sign,easy,tabl,nice

AUTHOR

Eric Chen, Apr 22 2015

STATUS

approved

A253854

Numbers b such that b^131072 + 1 is prime.

+0
22

1, 62722, 130816, 228188, 386892, 572186, 689186, 909548, 1063730, 1176694, 1361244, 1372930, 1560730, 1660830, 1717162, 1722230, 1766192, 1955556, 2194180, 2280466, 2639850, 3450080, 3615210, 3814944, 4085818, 4329134, 4893072, 4974408, 5326454, 5400728, 5471814

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Base values b yielding a generalized Fermat prime b^(2^k)+1 for k=17.

The first member exceeding 10^((10^6-1)/2^17) is known to be 42654182. - Jeppe Stig Nielsen, Jan 30 2016

LINKS

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..515 (from a massive computation by PrimeGrid).

C. K. Caldwell, 1560730^131072+1, The Largest Known Primes

C. K. Caldwell, Search result %^131072+1.

J. S. S. Nielsen, Generalized Fermat Primes sorted by base (see table at the bottom of the page)

PrimeGrid, PrimeGrid Primes - (GFN131072) Generalized Fermat Prime Search n=131072

PrimeGrid, Generalized Fermat statistics

CROSSREFS

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

KEYWORD

nonn,hard

AUTHOR

Felix Fröhlich, Jan 17 2015

EXTENSIONS

Missing term a(8) inserted by Jeppe Stig Nielsen, Jul 02 2015

a(13) from Felix Fröhlich, Nov 01 2015

a(14)-a(20) from Jeppe Stig Nielsen, Jan 30 2016

a(21)-a(31) from Jeppe Stig Nielsen, Sep 06 2017

a(1) = 1 inserted by Jeppe Stig Nielsen, Sep 10 2018

STATUS

approved

A250177

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

+0
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

MATHEMATICA

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

PROG

(PARI) {is(n)=isprime(polcyclo(21, n))};

for(n=1, 100, if(is(n)==1, print1(n, ", "), 0)) \\ G. C. Greubel, Apr 14 2018

CROSSREFS

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).

KEYWORD

nonn

AUTHOR

Eric Chen, Dec 24 2014

STATUS

approved

A250176

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

+0
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

MATHEMATICA

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

PROG

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

CROSSREFS

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).

KEYWORD

nonn

AUTHOR

Eric Chen, Dec 24 2014

EXTENSIONS

More terms from Vincenzo Librandi, Jan 16 2015

STATUS

approved

A250175

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

+0
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

MATHEMATICA

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

PROG

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

CROSSREFS

KEYWORD

nonn

AUTHOR

Eric Chen, Dec 24 2014

EXTENSIONS

More terms from Vincenzo Librandi, Jan 16 2015

STATUS

approved

A251597

Numbers b such that b^65536 + 1 is prime.

+0
23

1, 48594, 108368, 141146, 189590, 255694, 291726, 292550, 357868, 440846, 544118, 549868, 671600, 843832, 857678, 1024390, 1057476, 1087540, 1266062, 1361846, 1374038, 1478036, 1483076, 1540550, 1828502, 1874512, 1927034, 1966374, 2019300, 2041898, 2056292

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Base values b yielding a generalized Fermat prime b^(2^k) + 1 for k=16.

First square member of sequence is 3934049284 = (A253854(1))^2. - Jeppe Stig Nielsen, Jun 29 2015

LINKS

Ray Chandler, Table of n, a(n) for n = 1..1604 (2..70 from Felix Fröhlich, 71..425 from Jeppe Stig Nielsen)

C. K. Caldwell, The Prime Database, Search Output.

Y. Gallot, Generalized Fermat Prime Search

Merfighters, Table of known GF primes b^n+1 where n(exponent) is at least 8192, Mersenne Wiki

J. S. S. Nielsen, Generalized Fermat Primes sorted by base (see table at the bottom of the page)