A246397 -id:A246397

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A246392

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

+10
41

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

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

OFFSET

1,1

COMMENTS

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

LINKS

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)

MAPLE

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Eric Chen, Nov 13 2014

STATUS

approved

A085398

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

+10
25

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

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

OFFSET

1,1

COMMENTS

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

LINKS

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

Wikipedia, Bunyakowsky conjecture

FORMULA

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).

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Don Reble, Jun 28 2003

STATUS

approved

A250181

Numbers n such that Phi_26(n) is prime, where Phi is the cyclotomic polynomial.

+10
18

2, 3, 21, 22, 23, 35, 39, 74, 80, 84, 89, 108, 114, 121, 126, 134, 152, 153, 171, 180, 195, 204, 230, 256, 263, 297, 321, 326, 336, 342, 345, 351, 368, 390, 393, 397, 398, 399, 413, 427, 439, 490, 525, 563, 566, 574, 591, 602, 609, 630, 641, 652, 657, 660, 667, 682, 685, 694, 708, 712, 716, 721

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

OFFSET

1,1

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

MAPLE

select(t -> isprime((t^13+1)/(t+1)), [$1..2000]); # Robert Israel, Jan 15 2015

MATHEMATICA

Select[Range[2000], PrimeQ[(#^13 + 1) / (# + 1)] &] (* Vincenzo Librandi, Jan 15 2015 *)

PROG

(Magma) [n: n in [1..2000]| IsPrime((n^13+1) div (n+1))]; // Vincenzo Librandi, Jan 15 2015

(PARI) is(n)=isprime(polcyclo(26, n)) \\ Charles R Greathouse IV, Sep 08 2015

CROSSREFS

Cf. A246397.

KEYWORD

nonn

AUTHOR

R. J. Mathar, Jan 09 2015

STATUS

approved

A250177

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

+10
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

A250175

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

+10
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

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

A250176

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

+10
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

KEYWORD

nonn

AUTHOR

Eric Chen, Dec 24 2014

EXTENSIONS

More terms from Vincenzo Librandi, Jan 16 2015

STATUS

approved

A253240

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

+10
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

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