A019321 -id:A019321 - cp's OEIS frontend

A085028 Number of prime factors of cyclotomic(n,3), which is A019321(n), the value of the n-th cyclotomic polynomial evaluated at x=3.

1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 2, 2, 2, 1, 2, 2, 1, 2, 1, 3, 2, 3, 2, 3, 2, 1, 3, 2, 1, 2, 2, 4, 1, 3, 3, 2, 2, 3, 1, 4, 3, 5, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 1, 2, 2, 1, 2, 3, 2, 3, 2, 2, 1, 1, 1, 4, 3, 3, 2, 3, 4, 3, 2, 3, 2, 4, 2, 2, 1, 3, 3, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 4

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A057958, number of prime factors of 3^n-1.

See references at A085021

Max Alekseyev, Table of n, a(n) for n = 1..690

omega(Phi(n,x)): A085021 (x=2), this sequence (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).

Cf. A019321, A057958, A059885, A143663, A218356, A074477.

Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 3]]][[2]], {n, 1, 100}]

A138933 Indices k such that A019321(k)=Phi[k](3) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

1, 3, 6, 7, 9, 10, 12, 13, 14, 15, 21, 24, 26, 33, 36, 40, 46, 60, 63, 70, 71, 72, 86, 103, 108, 130, 132, 143, 145, 154, 161, 236, 255, 261, 276, 279, 287, 304, 364, 430, 464, 513, 528, 541, 562, 665, 672, 680, 707, 718, 747, 760, 782, 828, 875, 892, 974, 984

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

Robert Price, Table of n, a(n) for n = 1..167
Yves Gallot, Cyclotomic polynomials and prime numbers
Index entries for cyclotomic polynomials, values at X

Cf. A019321, A072226, A138920-A138940.

Select[Range[1000], PrimeQ[Cyclotomic[#, 3]] &]

for( i=1,999, ispseudoprime( polcyclo(i,3)) && print1( i","))

A368424 Numbers k such that gcd(A019320(k), A019321(k)) > 1.

Original entry on oeis.org

4, 11, 18, 20, 23, 28, 35, 43, 48, 52, 83, 95, 100, 119, 131, 138, 148, 155, 162, 166, 172, 179, 191, 196, 204, 210, 214, 239, 251, 253, 268, 292, 299, 300, 316, 323, 342, 359, 371, 378, 388, 419, 431, 443, 460, 463, 491, 500, 508, 515, 537, 556, 564, 575

Offset: 1

Tomohiro Yamada, Dec 24 2023

nonn

The corresponding greatest common divisors are given in A368425.

			a(1) = 4 since A019320(4) = 5 and A019321(4) = 10.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A019320, A019321, A191609 (prime factors of such gcds), A368425.

select(k -> igcd(numtheory:-cyclotomic(k,2),
numtheory:-cyclotomic(k,3)) > 1, [$1..1000]); # Robert Israel, Dec 26 2023

Select[Range[600],GCD[Cyclotomic[#,2],Cyclotomic[#,3]]>1&] (* Stefano Spezia, Dec 26 2023 *)

for(n=1,1000,if(gcd(polcyclo(n,2),polcyclo(n,3))>1,print1(n,", ")))

A019325 Cyclotomic polynomials at x=7.

Original entry on oeis.org

7, 6, 8, 57, 50, 2801, 43, 137257, 2402, 117993, 2101, 329554457, 2353, 16148168401, 102943, 4956001, 5764802, 38771752331201, 117307, 1899815864228857, 5649505, 11898664849, 247165843, 4561457890013486057, 5762401, 79797014141614001, 12111126301, 1628413638264057

Offset: 0

Simon Plouffe

Sequence has a(0) = x; see comments in A020501.

Robert Israel, Table of n, a(n) for n = 0..1186
Index entries for cyclotomic polynomials, values at X

Cf. A019320, A019321, A019322, A019323, A019324.

with(numtheory,cyclotomic); f := n->subs(x=7,cyclotomic(n,x)); seq(f(i),i=0..64);

Join[{7}, Cyclotomic[Range[50], 7]] (* Paolo Xausa, Feb 26 2024 *)

a(n) = if(n==0, 7, polcyclo(n, 7)); \\ Michel Marcus, Dec 16 2017

More terms from Michel Marcus, Dec 17 2017

A093107 Numbers n such that the Zsigmondy number Zs(n,3,1) differs from the n-th cyclotomic polynomial evaluated at 3.

Original entry on oeis.org

2, 4, 8, 16, 20, 32, 39, 42, 55, 64, 100, 128, 253, 256, 272, 294, 328, 342, 500, 507, 512, 605, 610, 666, 812, 876, 930, 1024, 1081, 1474, 1711, 1806, 2048, 2058, 2485, 2500, 2756, 2943, 3088, 3403, 3502, 4096, 4624, 4656, 5671, 5819, 6162, 6498, 6591, 6655

Offset: 1

Ralf Stephan, Mar 20 2004

nonn

A019321(n) does not equal A064079(n).

Cf. A093106, A093108, A093109.

Select[Range[10000], GCD[#, Cyclotomic[#, 3]]!=1 &] (* Emmanuel Vantieghem, Nov 13 2016 *)

More terms from Vladeta Jovovic, Apr 02 2004

Definition corrected by Jerry Metzger, Nov 04 2009

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)

A368425 The corresponding greatest common divisors to A368424(n).

Original entry on oeis.org

5, 23, 19, 5, 47, 29, 71, 431, 97, 53, 167, 191, 505, 239, 263, 139, 149, 311, 163, 499, 173, 359, 383, 197, 409, 211, 643, 479, 503, 23, 269, 293, 599, 1201, 317, 647, 19, 719, 743, 379, 389, 839, 863, 887, 461, 11113, 983, 5, 509, 1031, 4297, 557, 1129

Offset: 1

Tomohiro Yamada, Dec 24 2023

			a(2) = 23 since gcd(A019320(A368424(2)), A019321(A368424(2))) = gcd(2047, 88573) = 23.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A019320, A019321, A191609 (primes dividing some term of this sequence), A368424.

subs(1=NULL, [seq(igcd(numtheory:-cyclotomic(n,2), numtheory:-cyclotomic(n,3)),n=1..1000)]); # Robert Israel, Dec 26 2023

Select[GCD[Cyclotomic[Range[600], 2], Cyclotomic[Range[600], 3]],#>1&] (* Stefano Spezia, Dec 26 2023 *)

for(n=1,1000,m=gcd(polcyclo(n,2),polcyclo(n,3));if(m>1,print1(m,", ")))

A211874 Primes of the form Phi_k(3), the k-th cyclotomic polynomial evaluated at 3.

Original entry on oeis.org

2, 7, 13, 61, 73, 547, 757, 1093, 4561, 6481, 368089, 398581, 530713, 797161, 42521761, 47763361, 2413941289, 23535794707, 282429005041, 374857981681, 144542918285300809, 150094634909578633, 13490012358249728401, 82064241848634269407

Offset: 1

Alexander Gruber, Feb 12 2013

nonn

Primes in A019321.

Cf. A138933, A072226, A153601.

Sort[Select[Cyclotomic[Range[1000], 3], PrimeQ]]

cp's OEIS Frontend

A085028 Number of prime factors of cyclotomic(n,3), which is A019321(n), the value of the n-th cyclotomic polynomial evaluated at x=3.

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A138933 Indices k such that A019321(k)=Phi[k](3) is prime, where Phi is a cyclotomic polynomial.

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A368424 Numbers k such that gcd(A019320(k), A019321(k)) > 1.

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A019325 Cyclotomic polynomials at x=7.

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A093107 Numbers n such that the Zsigmondy number Zs(n,3,1) differs from the n-th cyclotomic polynomial evaluated at 3.

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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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A368425 The corresponding greatest common divisors to A368424(n).

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A211874 Primes of the form Phi_k(3), the k-th cyclotomic polynomial evaluated at 3.

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