A019331 -id:A019331 - cp's OEIS frontend

A019326 Cyclotomic polynomials at x=8.

8, 7, 9, 73, 65, 4681, 57, 299593, 4097, 262657, 3641, 1227133513, 4033, 78536544841, 233017, 14709241, 16777217, 321685687669321, 261633, 20587884010836553, 16519105, 60247241209, 954437177, 84327972908386521673, 16773121, 1152956690052710401, 61083979321

Offset: 0

Simon Plouffe

nonn

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for cyclotomic polynomials, values at X

Cf. A020500 (x = 1), A019320-A019331 (x = 2..13).

with(numtheory,cyclotomic); f := n->subs(x=8,cyclotomic(n,x)); seq(f(i),i=0..64);

Join[{8}, Cyclotomic[Range[50], 8]] (* Paolo Xausa, Feb 26 2024 *)

a(n) = if (n==0, 8, polcyclo(n, 8)); \\ Michel Marcus, Aug 07 2021

from sympy.polys.specialpolys import cyclotomic_poly
def a(n): return 8 if n == 0 else cyclotomic_poly(n, x=8)
print([a(n) for n in range(27)]) # Michael S. Branicky, Aug 07 2021

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)

A241039 Cyclotomic(n,2048).

Original entry on oeis.org

1, 2047, 2049, 4196353, 4194305, 17600780175361, 4192257, 73823022692637345793, 17592186044417, 73786976303428141057, 17583600302081, 1298708349570020393652962442872833, 17592181850113

Offset: 0

T. D. Noe, Apr 15 2014

nonn

Are all terms composite? At least the first 10000 terms are.

T. D. Noe, Table of n, a(n) for n = 0..300

Cf. A019320-A019331 (cyclotomic polynomials evaluated at 2..13).
Cf. A020500-A020513 (cyclotomic polynomials evaluated at 1, -2..-13, -1).
Cf. A117544 (least k such that cyclotomic(n,k) is prime).
Cf. A117545 (least k such that cyclotomic(k,n) is prime).

Mathematica
```
Table[Cyclotomic[k, 2048], {k, 0, 20}]
```

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A019326 Cyclotomic polynomials at x=8.

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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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A241039 Cyclotomic(n,2048).

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