A046887 -id:A046887 - cp's OEIS frontend

A013594 Smallest order of cyclotomic polynomial containing n or -n as a coefficient.

0, 105, 385, 1365, 1785, 2805, 3135, 6545, 6545, 10465, 10465, 10465, 10465, 10465, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 15015, 11305, 17255, 17255, 20615, 20615, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565

Offset: 1

N. J. A. Sloane

This sequence is infinite - see the Lang reference.

An alternative version would start with 1 rather than 0.

			a(2)=105 because cyclotomic(105) contains "-2" as coefficient, but for n < 105 cyclotomic(n) does not contain 2 or -2.
x^105 - 1 = ( - 1 + x)(1 + x + x^2)(1 + x + x^2 + x^3 + x^4)(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)(1 - x + x^3 - x^4 + x^5 - x^7 + x^8)(1 - x + x^3 - x^4 + x^6 - x^8 + x^9 - x^11 + x^12)(1 - x + x^5 - x^6 + x^7 - x^8 + x^10 - x^11 + x^12 - x^13 + x^14 - x^16 + x^17 - x^18 + x^19 - x^23 + x^24)(1 + x + x^2 - x^5 - x^6 - 2x^7 - x^8 - x^9 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 - x^20 - x^22 - x^24 - x^26 - x^28 + x^31 + x^32 + x^33 + x^34 + x^35 + x^36 - x^39 - x^40 - 2x^41 - x^42 - x^43 + x^46 + x^47 + x^48)

Bateman, C. Pomerance and R. C. Vaughan, Colloq. Math. Soc. Janos Bolyai, 34 (1984), 171-202.
S. Lang, Algebra: 3rd edition, Addison-Wesley, 1993, p. 281.
Maier, Prog. Math. 85 (Birkhaueser), 1990, 349-366.
Maier, Prog. Math. 139 (Birkhaueser) 1996, 633-638.

T. D. Noe, Table of n, a(n) for n = 1..1000
A. Arnold and M. Monagan, Calculating cyclotomic polynomials of very large height.
P. Erdős and R. C. Vaughan, Bounds for the r-th coefficients of cyclotomic polynomials, J. London Math. Soc. (2) 8 (1974), 393-400 (MR50 #9835; Zentralblatt 295.10014).
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
H. Maier, Cyclotomic polynomials with large coefficients, Acta Arith. 64 (1993), 227-235.
H. Maier, Cyclotomic polynomials whose orders contain many prime factors, Period. Math. Hungar. 43 (2001), 155-169.
H. L. Montgomery and R. C. Vaughan, The order of magnitude of the mth coefficients of cyclotomic polynomials, Glasgow Math. J. 27 (1985), 143-159.
R. C. Vaughan, Bounds for the coefficients of cyclotomic polynomials, Michigan Math. J. 21 (1974), 289-295 (1975).
M. Wallner, Lattice Path Combinatorics, Diplomarbeit, Institut für Diskrete Mathematik und Geometrie der Technischen Universität Wien, 2013.
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Cf. A046887, A013595, A013596, A063696, A063698, A134518, A137979.

Table[Position[Table[Max[Abs[Flatten[CoefficientList[Transpose[FactorList[x^i - 1]][[1]], x]]]], {i, 1, 10000}], j][[1]], {j, 1, 10}] (* Ian Miller, Feb 25 2008 *)

nm=6545; m=0; forstep(n=1, nm, 2, if(issquarefree(n), p=polcyclo(n); o=poldegree(p); for(k=0, o, a=abs(polcoeff(p, k)); if(a>m, m=a; print([m, n, factor(n)])))))

More terms from Eric W. Weisstein

Further terms from T. D. Noe, Oct 29 2007

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

A136418 Smallest order of the cyclotomic polynomial whose maximal coefficient in absolute value is n.

Original entry on oeis.org

0, 105, 385, 1365, 1785, 2805, 3135, 10353, 6545, 12155, 21385, 11165, 21505, 10465, 16555, 19285, 37961, 35105, 18445, 24395, 23205, 53669, 11305, 28595, 17255, 36465, 20615, 42315, 123585, 31535, 49335, 39585, 61295, 35805, 72709, 54285

Offset: 1

Robert G. Wilson v, Mar 31 2008

nonn

This differs from A013594.
For squarefree k, are there an infinite number of cyclotomic polynomials Phi(k,x) of height n? This is true for n=1 because it is known that there are an infinite number of flat cyclotomic polynomials with k the product of three distinct primes. See A117223. - T. D. Noe, Apr 22 2008
There are an infinite number of cyclotomic polynomials of height n if the following generalization of Kaplan's theorem 2 is true: Let N be the product of distinct odd primes and let p be one of those primes. Let q any prime such that q = p (mod N/p), then the height of Phi(Nq/p,x) is the same as the height of Phi(N,x). By Dirichlet's theorem, there are an infinite number of primes q. [From T. D. Noe, Apr 13 2010]

T. D. Noe, Table of n, a(n) for n=1..1000
Nathan Kaplan, Flat cyclotomic polynomials of order three, J. Number Theory 127 (2007), 118-126.

Cf. A013594, A046887, A134518.

f[n_] := f[n] = Max@ Abs@ CoefficientList[ Cyclotomic[n, x], x]; Do[ f@n, {n, 100000}]; t = Array[f, 31000]; Table[ Position[t, n, 1, 1], {n, 25}]//Flatten

More terms from T. D. Noe, Apr 22 2008

Showing 1-5 of 5 results.

cp's OEIS Frontend

A013594 Smallest order of cyclotomic polynomial containing n or -n as a coefficient.

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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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A136418 Smallest order of the cyclotomic polynomial whose maximal coefficient in absolute value is n.

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