A164989 -id:A164989 - cp's OEIS frontend

A246392 Numbers n such that Phi(10, n) is prime, where Phi is the cyclotomic polynomial.

2, 3, 5, 10, 11, 12, 16, 20, 21, 22, 33, 37, 38, 43, 47, 48, 55, 71, 75, 76, 80, 81, 111, 121, 126, 131, 133, 135, 136, 141, 155, 157, 158, 165, 176, 177, 180, 203, 223, 242, 245, 251, 253, 256, 257, 258, 265, 268, 276, 286, 290, 297, 307, 322, 323, 342, 361, 363, 366, 375, 377, 385, 388, 396, 411

Offset: 1

Eric Chen, Nov 13 2014

nonn

Numbers n such that (n^5+1)/(n+1) is prime, or numbers n such that A060884(n) is prime.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 893 terms from Robert Price)
OEIS Wiki, Cyclotomic Polynomials at x=sigma(n)

Cf. A008864 (1), A006093 (2), A002384 (3), A005574 (4), A049409 (5), A055494 (6), A100330 (7), A000068 (8), A153439 (9), this sequence (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).

Cf. A060884, A085398, A259257.

[n: n in [1..500]| IsPrime((n^5+1) div (n+1))]; // Vincenzo Librandi, Nov 14 2014

A246392:=n->`if`(isprime((n^5+1)/(n+1)),n,NULL): seq(A246392(n), n=1..500); # Wesley Ivan Hurt, Nov 15 2014

Select[Range[700], PrimeQ[(#^5 + 1) / (# + 1)] &] (* Vincenzo Librandi, Nov 14 2014 *)

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

A085398 Let Cn(x) be the n-th cyclotomic polynomial; a(n) is the least k>1 such that Cn(k) is prime.

Original entry on oeis.org

3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 6, 2, 4, 3, 2, 10, 2, 22, 2, 2, 4, 6, 2, 2, 2, 2, 2, 14, 3, 61, 2, 10, 2, 14, 2, 15, 25, 11, 2, 5, 5, 2, 6, 30, 11, 24, 7, 7, 2, 5, 7, 19, 3, 2, 2, 3, 30, 2, 9, 46, 85, 2, 3, 3, 3, 11, 16, 59, 7, 2, 2, 22, 2, 21, 61, 41, 7, 2, 2, 8, 5, 2, 2

Offset: 1

Don Reble, Jun 28 2003

nonn

Conjecture: a(n) is defined for all n. - Eric Chen, Nov 14 2014

Existence of a(n) is implied by Bunyakovsky's conjecture. - Robert Israel, Nov 13 2014

			a(11) = 5 because C11(k) is composite for k = 2, 3, 4 and prime for k = 5.
a(37) = 61 because C37(k) is composite for k = 2, 3, 4, ..., 60 and prime for k = 61.

Jinyuan Wang, Table of n, a(n) for n = 1..5000 (terms 1..1500 from Eric Chen)
Wikipedia, Bunyakowsky conjecture

Cf. A117544, A066180, A085399, A103795, A056993, A153438, A246119, A246120, A246121, A206418, A205506, A181980.

Cf. A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A006314, A217071, A164989, A217072, A217073, A153440, A217074, A217075, A006313, A097475.

f:= proc(n) local k;
for k from 2 do if isprime(numtheory:-cyclotomic(n,k)) then return k fi od
end proc:
seq(f(n), n = 1 .. 100); # Robert Israel, Nov 13 2014

Table[k = 2; While[!PrimeQ[Cyclotomic[n, k]], k++]; k, {n, 300}] (* Eric Chen, Nov 14 2014 *)

a(n) = k=2; while(!isprime(polcyclo(n, k)), k++); k; \\ Michel Marcus, Nov 13 2014

a(A072226(n)) = 2. - Eric Chen, Nov 14 2014
a(n) = A117544(n) except when n is a prime power, since if n is a prime power, then A117544(n) = 1. - Eric Chen, Nov 14 2014
a(prime(n)) = A066180(n), a(2*prime(n)) = A103795(n), a(2^n) = A056993(n-1), a(3^n) = A153438(n-1), a(2*3^n) = A246120(n-1), a(3*2^n) = A246119(n-1), a(6^n) = A246121(n-1), a(5^n) = A206418(n-1), a(6*A003586(n)) = A205506(n), a(10*A003592(n)) = A181980(n).

A153439 Numbers k such that k^6 + k^3 + 1 is prime.

Original entry on oeis.org

1, 2, 3, 8, 11, 20, 21, 26, 30, 50, 51, 56, 60, 78, 98, 102, 117, 129, 134, 146, 159, 171, 186, 189, 191, 198, 200, 209, 210, 212, 222, 240, 249, 267, 269, 278, 279, 299, 300, 333, 344, 363, 383, 390, 398, 399, 425, 429, 438, 444, 450, 458, 504, 509, 533, 540

Offset: 1

Pierre CAMI, Dec 26 2008

nonn

n^6+n^3+1 is the cyclotomic polynomial Phi(9,n). Damianou, p. 4, claims that there is strong experimental evidence that k^6 + k^3 + 1 is prime for infinitely many values of k. - Jonathan Vos Post, Jan 06 2011 [Comment corrected by Jon E. Schoenfield, Jan 22 2018]

Pierre CAMI, Table of n, a(n) for n = 1..37835
Pantelis A. Damianou, On prime values of cyclotomic polynomials, arXiv:1101.1152 [math.NT], 2011.

Cf. A153438, A164989.

[n: n in [0..500] | IsPrime(n^3*(n^3+1)+1)] // Vincenzo Librandi, Nov 26 2010

Select[Range@ 600, PrimeQ[#^6 + #^3 + 1] &] (* Michael De Vlieger, Jan 22 2018 *)

is(n)=prime(n^6+n^3+1) \\ Charles R Greathouse IV, Feb 17 2017

Some unclear comments deleted by N. J. A. Sloane, Sep 06 2009

More terms from Vincenzo Librandi, Mar 25 2010

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

A175170 Primes of the form k^6 - k^3 + 1.

Original entry on oeis.org

46441, 117307, 530713, 5488957657, 13841169553, 24794753833, 82653662521, 151333837273, 282429005041, 464403405313, 782756904961, 999999000001, 1418517921241, 2699552509993, 2985982272001, 3138426605161, 4001502141001, 28722895031251, 47045874141001, 56693904845761, 78672332016307

Offset: 1

Vincenzo Librandi, Dec 03 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A060891, A164989.

[ a: n in [0..250] | IsPrime(a) where a is n^3*(n^3-1)+1 ];

Select[Table[n^6-n^3+1,{n,0,2000}],PrimeQ] (* Vincenzo Librandi, Jul 26 2012 *)

A000040 INTERSECT A060891.

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

A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

Original entry on oeis.org

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

Offset: 0

Eric Chen, Apr 22 2015

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)).

			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
...

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

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).

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

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))

T(m, n) = Phi_m(n)

A175205 Primes p such that p^6-p^3+1 is also prime.

Original entry on oeis.org

7, 73, 271, 367, 379, 397, 739, 823, 859, 1291, 1297, 1627, 1747, 2083, 2203, 2221, 2269, 2311, 3067, 3079, 3109, 3163, 3463, 3607, 3733, 3907, 4093, 4339, 4603, 4933, 5077, 5431, 5527, 5821, 5869, 6091, 6301, 6373, 6421, 6529, 6883, 7927, 7951, 8011, 8467, 8689, 8737

Offset: 1

Vincenzo Librandi, Dec 03 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A060891, A175170, A000040, A164989.

Subsequence of A002476.

[p: p in PrimesUpTo(9000) | IsPrime(p^6-p^3+1)]; // Vincenzo Librandi, Apr 16 2013

Select[Prime[Range[2000]], PrimeQ[#^6 -  #^3 + 1]&] (* Vincenzo Librandi, Apr 16 2013 *)

forprime(p=2,10000,if(isprime(p^6-p^3+1),print1(p,", ")))

A000040 INTERSECT A164989.

cp's OEIS Frontend

A246392 Numbers n such that Phi(10, n) is prime, where Phi is the cyclotomic polynomial.

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A085398 Let Cn(x) be the n-th cyclotomic polynomial; a(n) is the least k>1 such that Cn(k) is prime.

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A153439 Numbers k such that k^6 + k^3 + 1 is 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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A175170 Primes of the form k^6 - k^3 + 1.

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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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