A262405 -id:A262405 - cp's OEIS frontend

A013595 Irregular triangle read by rows: coefficients of cyclotomic polynomial Phi_n(x) (exponents in increasing order).

0, 1, -1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, -1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 0, 1, -1, 1, 0, -1, 1

Offset: 0

N. J. A. Sloane

We follow Maple in defining Phi_0 to be x; it could equally well be taken to be 1.
From Wolfdieter Lang, Oct 29 2013: (Start)
The length of row n  >= 1 of this  table is phi(n) + 1 = A000010(n) + 1. Row n = 0 has here length 2.
Phi_n(x) is the minimal polynomial of omega_n := exp(i*2*Pi/n) over the rationals. Namely, Phi_n(x) = Product_{k=0..n-1, gcd(k,n)=1} (x - (omega_n)^k). See the Graham et al. reference, 4.50 a, pp. 149, 506.
Phi_n(x) = Product_{d|n} (x^d - 1)^(mu(n/d)) with the Moebius function mu(n) = A008683(n), n >= 1. See the Graham et al. reference, 4.50 b, pp. 149, 506.
Phi_n(x) = Phi_{rad(n)}(x^(n/rad(n))), n >= 2, with rad(n) = A007947(n), the squarefree kernel of n. Proof from the preceding formula, where only squarefree n/d (A005117) from the set of divisors of n enter, by mapping each factor (numerator or denominator) of the left hand side to one of the right hand side and vice versa.
(End)
Each row can be considered as the last column of the companion matrix of the cyclotomic polynomial: A000010(n) is the size of such a square matrix, last column has opposite signs and the last term (before last term of each row in A013595) equal to A008683(n). - Eric Desbiaux, Dec 14 2015

			Phi_0 = x; Phi_1 = x - 1; Phi_2 = x + 1; Phi_3 = x^2 + x + 1; Phi_4 = x^2 + 1; ...
From _Wolfdieter Lang_, Oct 29 2013: (Start)
The irregular triangle a(n,m) begins:
n\m 0  1  2  3  4  5  6  7  8  9 10 11 12 ...
0:  0  1
1: -1  1
2:  1  1
3:  1  1  1
4:  1  0  1
5:  1  1  1  1  1
6:  1 -1  1
7:  1  1  1  1  1  1  1
8:  1  0  0  0  1
9:  1  0  0  1  0  0  1
10: 1 -1  1 -1  1
11: 1  1  1  1  1  1  1  1  1  1  1
12: 1  0 -1  0  1
13: 1  1  1  1  1  1  1  1  1  1  1  1  1
14: 1 -1  1 -1  1 -1  1
15: 1 -1  0  1 -1  1  0 -1  1
...
Phi_15(x) = (x^1 - 1)*((x^3 - 1)^(-1))*((x^5 - 1)^(-1))*(x^15 - 1) because mu(15) = mu(1) = +1 and mu(3) = mu(5) = -1. Hence Phi_15(x) = 1 - x + x^3 - x^4 + x^5 - x^7 + x^8, giving row n = 15.
Example for the reduction via the squarefree kernel: Phi_12(x) = Phi_6(x^(12/6)) = Phi_6(x^2). By the formula with the Mobius function Phi_6(x) = Phi_2(x^3)/Phi_2(x) = 1 - x + x^2 and with x -> x^2 this becomes Phi_12(x) = 1 - x^2 + x^4.
(End)

E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, 1968; see p. 90.
Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, p. 325.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1991, p. 137.
K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, 1982, p. 194.

Robert Israel, Table of n, a(n) for n = 0..10000
Emma Lehmer, On the magnitude of the coefficients of the cyclotomic polynomial, Bull. Amer. Math. Soc. 42 (1936), 389-392.
Rakshith Rajashekar, Marco Di Renzo, K.V.S. Hari, L. Hanzo, A generalised transmit and receive diversity condition for feedback assisted MIMO systems: theory & applications in full-duplex spatial modulation, 2017.
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial.
Wikipedia, Cyclotomic Polynomial.

Cf. A013596, A020500 (row sums, n >= 1), A020513 (alternating row sums).
For record coefficients see A160340, A262404, A262405, A278567.
Column m=1 is A157657.

N:= 100:  # to get coefficients up to cyclotomic(N,x)
with(numtheory):
for n from 0 to N do
  C:= cyclotomic(n,x);
  L[n]:= seq(coeff(C,x,i),i=0..degree(C));
od:
A:= [seq](L[n],n=0..N): # note that A013595(n) = A[n+1]
# Robert Israel, Apr 17 2014

Table[CoefficientList[x^KroneckerDelta[n] Cyclotomic[n, x], x], {n, 0, 15}] // Flatten (* Peter Luschny, Dec 27 2016 *)

row(n) = if (n==0, p=x, p = polcyclo(n)); Vecrev(p); \\ Michel Marcus, Dec 14 2015

a(n,m) = [x^m] Phi_n(x), n >= 0, 0 <= m <= phi(n), with phi(n) = A000010(n). - Wolfdieter Lang, Oct 29 2013

Maple program corrected by Robert Israel, Apr 17 2014

A160340 Indices of records in heights of cyclotomic polynomials (A160338).

Original entry on oeis.org

1, 105, 385, 1365, 1785, 2805, 3135, 6545, 10465, 11305, 17255, 20615, 26565, 40755, 106743, 171717, 255255, 279565, 327845, 707455, 886445, 983535, 1181895, 1752465, 3949491, 8070699, 10163195, 13441645, 15069565, 30489585, 37495115, 40324935

Offset: 1

Max Alekseyev, May 13 2009

nonn

m is in this sequence if A160338(k) < A160338(m) for all k

Max Alekseyev, Table of n, a(n) for n = 1..42
John Abbott and Nico Mexis, Cyclotomic Factors and LRS-Degeneracy, arXiv:2403.08751 [math.AC], 2024. See p. 12.
Andrew Arnold and Michael Monagan, Calculating cyclotomic polynomials of very large height.
Lola Thompson, Cyclotomic statistics, Univ. Utrecht (Netherlands, 2024). See pp. 6, 14.

Subsequence of A013594 and A046887.

See also A013595, A262404, A262405, A278567.

r = 0; Do[If[# > r, r = #; Print[n]] &@ Max@ Abs@ CoefficientList[Cyclotomic[n, x], x], {n, 10^4}] (* Michael De Vlieger, May 20 2024 *)

print1(r=1); for(n=2,1e4, t=vecmax(abs(Vec(polcyclo(n)))); if(t>r, r=t; print1(", "n))) \\ Charles R Greathouse IV, Jun 28 2012

A278567 Maximal coefficient (in absolute value) of cyclotomic polynomial C(N,x), where N = n-th number which is a product of exactly three distinct primes = A007304(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2

Offset: 1

N. J. A. Sloane, Nov 26 2016

nonn

E. Lehmer (1936) shows that this sequence is unbounded.

			The first 2 occurs in the famous C(105,x), which is x^48+x^47+x^46-x^43-x^42-2*x^41-x^40-x^39+x^36+x^35+x^34+x^33+x^32+x^31-x^28-x^26-x^24-x^22-x^20+x^17+x^16+x^15+x^14+x^13+x^12-x^9-x^8-2*x^7-x^6-x^5+x^2+x+1.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Emma Lehmer, On the magnitude of the coefficients of the cyclotomic polynomial, Bull. Amer. Math. Soc. 42 (1936), 389-392.

See A278571 for smallest m such that a(m) = n.
Cf. A007304, A013595, A160340, A262404, A262405.
See A278570 for another version.

with(numtheory):
b:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, b(n-1)) while
      bigomega(k)<>3 or nops(factorset(k))<>3 do od; k
    end:
a:= n-> max(map(abs, [coeffs(cyclotomic(b(n), x))])):
seq(a(n), n=1..120);  # Alois P. Heinz, Nov 26 2016

f[n_] := Max[ Abs[ CoefficientList[ Cyclotomic[n, x], x]]]; t = Take[ Sort@ Flatten@ Table[Prime@i Prime@j Prime@k, {i, 3, 35}, {j, 2, i -1}, {k, j -1}], 105]; f@# & /@ t (* Robert G. Wilson v, Dec 09 2016 *)

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, cyclotomic_poly
def A278567(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1),1) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return max(int(abs(x[1][0][0])) for x in cyclotomic_poly(bisection(f)).as_terms()[0]) # Chai Wah Wu, Aug 31 2024

A262404 Least k such that the k-th cyclotomic polynomial has n as a coefficient.

Original entry on oeis.org

4, 1, 165, 595, 1785, 1785, 2805, 3135, 6545, 6545, 10465, 10465, 10465, 10465, 10465, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 15015, 11305, 20615, 17255, 20615, 20615, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565

Offset: 0

Charles R Greathouse IV, Sep 21 2015

Suzuki proves that a(n) exists for each n. Vaughan proves that there are infinitely many k with a(n) = k and n > exp(exp(log 2 * log k/log log k)).

			Phi(165) = x^80 + x^79 + x^78 - x^75 - x^74 - x^73 - x^69 - x^68 - x^67 + x^65 + 2x^64 + 2x^63 + x^62 - x^60 - x^59 - x^58 - x^54 - x^53 - x^52 + x^50 + 2x^49 + 2x^48 + 2x^47 + x^46 - x^44 - x^43 - x^42 - x^41 - x^40 - x^39 - x^38 - x^37 - x^36 + x^34 + 2x^33 + 2x^32 + 2x^31 + x^30 - x^28 - x^27 - x^26 - x^22 - x^21 - x^20 + x^18 + 2x^17 + 2x^16 + x^15 - x^13 - x^12 - x^11 - x^7 - x^6 - x^5 + x^2 + x + 1, with 2 as the coefficient of x^16 (among others), and this is the least k for which 2 appears, so a(2) = 165.

Robert Israel, Table of n, a(n) for n = 0..1000
Jiro Suzuki, On coefficients of cyclotomic polynomials, Proc. Japan Acad. Ser. A Math. Sci. 63:7 (1987), pp. 279-280.
R. C. Vaughan, Bounds for the coefficients of cyclotomic polynomials, Michigan Math. J. 21 (1974), 289-295 (1975).

Cf. A013594, A013595, A046887, A262405, A160340, A278567.

N:= 40: count:= 0: A:= Array(0..N): A[0]:= 4:
for k from 1 while count < N do
  S:= select(t -> t::posint and t <= N and A[t] = 0, {coeffs(numtheory:-cyclotomic(k,x),x)}):
  if S <> {} then
    A[convert(S,list)]:= k;
    count:= count + nops(S);
  fi
od:
convert(A,list); # Robert Israel, Dec 23 2018

Table[k = 1; While[! MemberQ[CoefficientList[Cyclotomic[k, x], x], n], k++]; k, {n, 0, 9}] (* Michael De Vlieger, Sep 29 2015 *)

a(n)=my(k,v);while(!setsearch(Set(Vec(polcyclo(k++))),n),);k

Corrected a(22); more terms from Seiichi Manyama, Dec 22 2018

A131672 a(n) is the smallest k >= 1 such that the expansion of the inverse of the k-th cyclotomic polynomial has n or -n as a coefficient, or -1 if no such k exists.

Original entry on oeis.org

1, 561, 1155, 2145, 3795, 5005, 5005, 8645, 8645, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 31395, 31395, 31395, 31395, 31395, 33495, 33495, 33495, 33495, 33495, 33495, 33495, 33495, 33495, 33495, 33495, 33495, 40755

Offset: 1

Jonathan Vos Post, Sep 12 2007

nonn

Moree's abstract: "Let Psi_n(x) be the monic polynomial having precisely all nonprimitive n-th roots of unity as its simple zeros. One has Psi_n(x) = (x^n-1)/Phi_n(x), with Phi_n(x) the n-th cyclotomic polynomial. The coefficients of Psi_n(x) are integers that like the coefficients of Phi_n(x) tend to be surprisingly small in absolute value, e.g. for n<561 all coefficients of Psi_n(x) are <= 1 in absolute value. We establish various properties of the coefficients of Psi_n(x).
From Jianing Song, May 26 2021: (Start)
The old name was "Arises in reciprocal cyclotomic polynomials".
Since 1/Phi_n(x) = -Psi_n(x) * (1 + x^n + x^(2n) + ...), the period length of coefficients in the expansion of 1/Phi_n(x) is n.
For n > 1, a(n) is odd, composite and squarefree.
Conjecture 1: a(n) > 0 exists for all n.
Conjecture 2: for every k > 2, there exists some positive integer m such that the coefficients in the expansion of 1/Phi_k(x) are -m, -(m-1), ..., m-1, m. If this is true, then for n > 1, a(n) is also the smallest k such that the expansion of 1/Phi_k(x) has both n and -n as coefficients, and a(n) is also the smallest k such that the expansion of 1/Phi_k(x) has a coefficient whose absolute value is greater than or equal to n. (End)
So far, all terms k > 1 have the property that k is squarefree with omega(k) > 2. - Robert G. Wilson v, Jun 09 2021

			The cyclotomic polynomial Phi_1(x) = 1-x (cf. A013595), so the inverse cyclotomic polynomial Psi_1(x) = 1 (cf. A306453), and so a(1) = 1. - _N. J. A. Sloane_, Jun 08 2021

Robert G. Wilson v, Table of n, a(n) for n = 1..341
Pieter Moree, Reciprocal cyclotomic polynomials, arXiv:0709.1570 [math.NT], Sep 11 2007, table 1 (computed by Yves Gallot), p. 13.

Cf. A013595, A306453, A333248, A262404, A262405.

a(n) = my(k=1); while((k%2==0) || (isprime(k)) || (!issquarefree(k)) || !setsearch(Set(abs(Vec((x^k-1)/polcyclo(k)))), n), k++); k \\ Jianing Song, May 26 2021

New name from Jianing Song, May 26 2021

Terms a(22) onward from Robert G. Wilson v, Jun 09 2021

Showing 1-5 of 5 results.

cp's OEIS Frontend

A013595 Irregular triangle read by rows: coefficients of cyclotomic polynomial Phi_n(x) (exponents in increasing order).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A160340 Indices of records in heights of cyclotomic polynomials (A160338).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A278567 Maximal coefficient (in absolute value) of cyclotomic polynomial C(N,x), where N = n-th number which is a product of exactly three distinct primes = A007304(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

A262404 Least k such that the k-th cyclotomic polynomial has n as a coefficient.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A131672 a(n) is the smallest k >= 1 such that the expansion of the inverse of the k-th cyclotomic polynomial has n or -n as a coefficient, or -1 if no such k exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions