A246392 -id:A246392 - cp's OEIS frontend

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

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)

A259257 Primes of the form n^4 - n^3 + n^2 - n + 1.

Original entry on oeis.org

11, 61, 521, 9091, 13421, 19141, 61681, 152381, 185641, 224071, 1151041, 1824841, 2031671, 3341101, 4778021, 5200081, 8987221, 25058741, 31224301, 32928901, 40454321, 42521761, 150451621, 212601841, 250062751, 292268861, 310565641, 329708341, 339604921

Offset: 1

Robert Price, Jun 22 2015

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. A060884, A246392.

[a: n in [0..200] | IsPrime(a) where a is n^4-n^3+n^2-n+1]; // Vincenzo Librandi, Jun 23 2015

Select[Table[Cyclotomic[10, n], {n, 0, 200}], PrimeQ]
Select[Table[n^4 - n^3 + n^2 - n + 1, {n, 200}], PrimeQ] (* Vincenzo Librandi, Jun 23 2015 *)

lista(nn) = for (n=1, nn, if (isprime(p=polcyclo(10, n)), print1(p, ", "))); \\ Michel Marcus, Jun 23 2015

a(n) = A246392(A060884(n)).

A250182 Numbers n such that Phi_28(n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

4, 5, 7, 13, 25, 33, 41, 63, 74, 80, 88, 94, 96, 104, 116, 144, 149, 151, 154, 165, 167, 174, 183, 191, 197, 208, 231, 241, 262, 268, 270, 290, 318, 319, 361, 368, 376, 390, 394, 412, 431, 434, 442, 464, 489, 492, 521, 529, 556, 568, 574, 575, 585, 589, 613, 629, 639, 654, 666, 667, 672, 683, 684

Offset: 1

R. J. Mathar, Jan 09 2015

nonn

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

Cf. A246392.

Select[Range[700], PrimeQ[Cyclotomic[28, #]] &] (* Vincenzo Librandi, Jan 17 2015 *)

isok(n) = isprime(polcyclo(28, n)); \\ Michel Marcus, Jan 17 2015

A250183 Numbers n such that Phi(30,n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 6, 7, 8, 9, 16, 18, 20, 26, 29, 31, 32, 33, 40, 41, 47, 57, 76, 82, 87, 88, 93, 101, 109, 120, 121, 125, 133, 140, 144, 162, 163, 175, 178, 183, 186, 191, 196, 215, 216, 218, 227, 233, 242, 253, 266, 267, 273, 276, 278, 304, 312, 317, 319, 328, 336, 374, 380

Offset: 1

R. J. Mathar, Jan 12 2015

nonn

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

Cf. A246392.

Select[Range[400], PrimeQ[Cyclotomic[30, #]] &] (* Vincenzo Librandi, Jan 17 2015 *)

isok(n) = isprime(polcyclo(30, n)); \\ Michel Marcus, Jan 17 2015

A250184 Numbers n such that Phi(33,n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 3, 13, 14, 18, 31, 44, 59, 62, 75, 80, 104, 109, 145, 185, 213, 273, 282, 309, 321, 337, 379, 399, 405, 411, 430, 452, 464, 470, 522, 535, 560, 566, 586, 593, 597, 653, 654, 688, 702, 704, 727, 728, 744, 746, 780, 805, 806, 816, 822, 829, 846, 856

Offset: 1

R. J. Mathar, Jan 12 2015

nonn

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

Cf. A246392.

Select[Range[900], PrimeQ[Cyclotomic[33, #]] &] (* Vincenzo Librandi, Jan 17 2015 *)

isok(n) = isprime(polcyclo(30, n)); \\ Michel Marcus, Jan 17 2015

A250186 Numbers n such that Phi(35,n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

14, 33, 39, 55, 112, 130, 132, 138, 168, 176, 179, 186, 189, 195, 210, 246, 259, 264, 310, 318, 346, 417, 431, 467, 478, 480, 534, 545, 564, 567, 661, 671, 741, 744, 749, 757, 786, 794, 804, 825, 851, 866, 911, 948, 955, 962, 976, 992, 1014, 1033, 1042, 1082, 1109, 1161, 1193, 1220, 1244, 1260, 1268, 1278, 1313, 1414, 1437

Offset: 1

R. J. Mathar, Jan 12 2015

nonn

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

Cf. A246392.

Select[Range[1500], PrimeQ[Cyclotomic[35, #]] &] (* Vincenzo Librandi, Jan 16 2015 *)

{is(n)=isprime(polcyclo(35,n))};
for(n=1,1000, if(is(n), print1(n, ", "))) \\ G. C. Greubel, May 20 2018

A250188 Numbers n such that Phi(39,n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

10, 23, 86, 240, 254, 284, 340, 369, 371, 378, 382, 407, 422, 448, 459, 574, 582, 613, 619, 667, 686, 703, 767, 769, 844, 851, 875, 881, 944, 987, 995, 1207, 1219, 1233, 1279, 1292, 1343, 1372, 1399, 1409, 1445, 1468, 1497, 1500, 1557, 1586, 1598, 1633, 1645, 1677, 1760, 1807, 1835

Offset: 1

R. J. Mathar, Jan 12 2015

nonn

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

Cf. A246392.

Select[Range[1900], PrimeQ[Cyclotomic[39, #]] &] (* Vincenzo Librandi, Jan 16 2015 *)

{is(n)=isprime(polcyclo(39,n))};
for(n=1,1000, if(is(n), print1(n, ", "))) \\ G. C. Greubel, May 20 2018

A250189 Numbers n such that Phi(40,n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 3, 4, 31, 37, 39, 46, 77, 98, 119, 124, 143, 144, 154, 169, 197, 205, 232, 245, 266, 291, 295, 297, 305, 308, 319, 332, 333, 413, 426, 431, 437, 454, 459, 472, 483, 513, 528, 531, 542, 579, 617, 619, 635, 647, 677, 724, 737, 748, 780, 806, 815, 819, 820, 840, 851, 858, 870, 875, 907, 920, 927, 940, 983, 1002, 1028

Offset: 1

R. J. Mathar, Jan 12 2015

nonn

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

Cf. A246392.

Select[Range[1200], PrimeQ[Cyclotomic[40, #]] &] (* Vincenzo Librandi, Jan 16 2015 *)

{is(n)=isprime(polcyclo(40,n))};
for(n=1,1000, if(is(n), print1(n, ", "))) \\ G. C. Greubel, May 20 2018

A250190 Numbers n such that Phi(42,n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 6, 8, 11, 23, 35, 41, 49, 58, 65, 85, 88, 97, 107, 111, 139, 144, 161, 170, 197, 214, 217, 223, 230, 238, 247, 274, 298, 301, 323, 382, 389, 393, 398, 403, 427, 445, 452, 473, 480, 497, 511, 561, 575, 595, 601, 604, 606, 615, 629, 651, 652, 680, 685, 702, 725, 762, 770, 774, 781, 805, 814, 912, 918, 942, 987, 1030

Offset: 1

R. J. Mathar, Jan 12 2015

nonn

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

Cf. A246392.

Select[Range[1100], PrimeQ[Cyclotomic[42, #]] &] (* Vincenzo Librandi, Jan 16 2015 *)

{is(n)=isprime(polcyclo(42,n))};
for(n=1,1000, if(is(n)==1, print1(n, ", "))) \\ G. C. Greubel, May 18 2018

cp's OEIS Frontend

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

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

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A259257 Primes of the form n^4 - n^3 + n^2 - n + 1.

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A250182 Numbers n such that Phi_28(n) is prime, where Phi is the cyclotomic polynomial.

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A250183 Numbers n such that Phi(30,n) is prime, where Phi is the cyclotomic polynomial.

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A250184 Numbers n such that Phi(33,n) is prime, where Phi is the cyclotomic polynomial.

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A250186 Numbers n such that Phi(35,n) is prime, where Phi is the cyclotomic polynomial.

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A250188 Numbers n such that Phi(39,n) is prime, where Phi is the cyclotomic polynomial.

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A250189 Numbers n such that Phi(40,n) is prime, where Phi is the cyclotomic polynomial.