A137979 -id:A137979 - 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

A013590 Numbers k such that Phi(k,x) is a cyclotomic polynomial containing a coefficient with an absolute value greater than one.

Original entry on oeis.org

105, 165, 195, 210, 255, 273, 285, 315, 330, 345, 357, 385, 390, 420, 429, 455, 495, 510, 525, 546, 555, 561, 570, 585, 595, 609, 615, 627, 630, 645, 660, 665, 690, 705, 714, 715, 735, 759, 765, 770, 777, 780, 795, 805, 819, 825, 840, 855

Offset: 1

Peter T. Wang (peterw(AT)cco.caltech.edu)

nonn

Previous name was: Orders of cyclotomic polynomials containing a coefficient with an absolute value greater than one.
First occurrence of A137979(n)=k is given in A013594.
From David A. Corneth, Apr 21 2018: (Start)
Terms are composite.
If k is a term of the sequence then so is k * m for m > 0.
Let a primitive term p of this sequence be a term of which no divisor is in the sequence. Then p is an odd squarefree number. (End)

Robert G. Wilson v, Table of n, a(n) for n = 1..1627

Cf. A013594, A117223, A117318, A137979.

Flat cyclotomic polynomial: A117223 (order 3), A117318 (order 4).

isA013590 := proc(n)
    numtheory[cyclotomic](n,x) ;
    {coeffs(%,x)} ;
    map(abs,%) ;
    if % minus {1}  = {} then
        false;
    else
        true;
    end if;
end proc:
for n from 1 do
    if isA013590(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 28 2016

S[ n_ ] := For[ j=1; t=0, j1 ]; If[ Length[ t ]!=0, Print[ j ] ] ]; S[ 856 ]
f[n_] := Max@ Abs@ CoefficientList[ Cyclotomic[n, x], x]; Select[ Range@ 1000, f@# > 1 &] (* Robert G. Wilson v *)
Select[Range[900],Max[Abs[CoefficientList[Cyclotomic[#,x],x]]]>1&] (* Harvey P. Dale, Mar 13 2013 *)

is(n)=for(k=0,n,if(abs(polcoeff(polcyclo(n),k))>1,return(n)));0
for(n=1,1000,if(is(n),print1(n,", "))) \\ Derek Orr, Apr 22 2015

Definition clarified by Harvey P. Dale, Mar 13 2013

New name from Michel Marcus, Apr 29 2018

A160338 Height (maximum absolute value of coefficients) of the n-th cyclotomic polynomial.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Max Alekseyev, May 13 2009

Different from A137979: first time these sequence disagree is at n=14235 with a(14235)=2 and A137979(14235)=3.

			a(4) = 1 because the 4th cyclotomic polynomial x^2 + 1 has height 1.

Max Alekseyev, Table of n, a(n) for n = 1..100000
Alexandre Kosyak, Pieter Moree, Efthymios Sofos and Bin Zhang, Cyclotomic polynomials with prescribed height and prime number theory, arXiv:1910.01039 [math.NT], 2019.
Emma Lehmer, On the magnitude of the coefficients of the cyclotomic polynomial, Bull. Amer. Math. Soc. 42 (1936), 389-392.
H. Maier, The coefficients of cyclotomic polynomials, Analytic number theory, Vol. 2 (1995), pp. 633-639, Progr. Math., 139.
Lola Thompson, Heights of divisors of x^n-1, arXiv:1111.5404 [math.NT], 2011.
R. C. Vaughan, Bounds for the coefficients of cyclotomic polynomials, Michigan Math. J. 21 (1974), 289-295 (1975).

Cf. A160339 (records), A160340 (indices of records), A160341.

Table[Max@Abs@CoefficientList[Cyclotomic[n,x],x],{n,1,105}] (* from Jean-François Alcover, Apr 02 2011 *)

PARI
```
a(n) = vecmax(abs(Vec(polcyclo(n))))
```

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

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A013590 Numbers k such that Phi(k,x) is a cyclotomic polynomial containing a coefficient with an absolute value greater than one.

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A160338 Height (maximum absolute value of coefficients) of the n-th cyclotomic polynomial.

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Mathematica

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