A278567 -id:A278567 - 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

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

A262405 Least k such that the k-th cyclotomic polynomial has -n as a coefficient.

Original entry on oeis.org

4, 1, 105, 385, 1365, 2145, 2805, 3135, 6545, 7917, 10465, 10465, 10465, 10465, 10465, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 15015, 17255, 17255, 17255, 20615, 25935, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565

Offset: 0

Charles R Greathouse IV, Sep 21 2015

nonn

Suzuki proves that a(n) exists for each n.

			Phi(105) = x^48 + x^47 + x^46 - x^43 - x^42 - 2x^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 - 2x^7 - x^6 - x^5 + x^2 + x + 1, with -2 as the coefficient of x^7 (among others), and this is the least k for which -2 appears, so a(2) = 105.

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, A262404, A160340, A278567.

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

More terms from Seiichi Manyama, Dec 22 2018

A278570 a(n) = maximum absolute value of coefficients in the cyclotomic polynomial C(N,x), where N = n-th number which a product of three distinct odd primes = A046389(n).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 27 2016

nonn

Don Reble, Posting to Sequence Fans Mailing List, Nov 26 2016

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A046389. See A278567 for a closely related sequence.

with(numtheory):
b:= proc(n) option remember; local k;
      for k from 2+`if`(n=1, 1, b(n-1)) by 2 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 27 2016

b[n_] := b[n] = (For[k = 2 + If[n == 1, 1, b[n-1]], PrimeOmega[k] != 3 || PrimeNu[k] != 3, k += 2]; k);
a[n_] :=  Max @ Abs @ CoefficientList[Cyclotomic[b[n], x], x];
Array[a, 120] (* Jean-François Alcover, Mar 28 2017, after Alois P. Heinz *)

A373196 Maximal coefficient (in absolute value) in the numerator of C({1..n},x).

Original entry on oeis.org

1, 1, 2, 17, 444, 66559954, 14648786369948422, 791540878703169050660325841979096789557779, 1918013047695258943191946313451491492494186620117241479813740479213857275772347178176158

Offset: 0

John Tyler Rascoe, Jun 28 2024

nonn

			C_x({1,2,3},x) = (-x^15 - 5*x^14 - 12*x^13 - 17*x^12 - 11*x^11 + 4*x^10 + 16*x^9 + 10*x^8 - 6*x^6)/(x^15 + 4*x^14 + 7*x^13 + 4*x^12 - 8*x^11 - 18*x^10 - 13*x^9 + 7*x^8 + 19*x^7 + 11*x^6 - 6*x^5 - 10*x^4 - 2*x^3 + 3*x^2 + 2*x - 1) with maximal coefficient abs(-17) in the numerator, so a(3) = 17.

Cf. A107429, A114536, A278567.

C_x(s)={my(g=if(#s <1,1, sum(i=1,#s, C_x(s[^i])*x^(s[i]))/(1-sum(i=1,#s, x^(s[i]))))); return(g)}
a(n)={vecmax(abs(Vec(numerator(C_x([1..n])))))}

C({s},x) = Sum_{i in {s}} (C({s}-{i},x)*x^i)/(1 - Sum_{i in {s}} (x^i)) with C({},x) = 1.

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A278571 Index of first occurrence of n in A278567.

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

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

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

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A262405 Least k such that the k-th cyclotomic polynomial has -n as a coefficient.

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A278570 a(n) = maximum absolute value of coefficients in the cyclotomic polynomial C(N,x), where N = n-th number which a product of three distinct odd primes = A046389(n).

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Maple

Mathematica

A373196 Maximal coefficient (in absolute value) in the numerator of C({1..n},x).

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PARI

Formula