A137979 Highest coefficient occurring in the factorization of x^n - 1 over the reals.

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

1,105

Comments

Based on a comment in Mathematica helpfile ref/Factor - Neat Examples.

The first factorization of x^n - 1 in which a 2 appears as a coefficient is for n=105.

Different from A160338, see comment there.

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Example

a(4) = 1 because x^4 - 1 = (x^2+1)(x+1)(x-1) and the highest coefficient of these three terms is 1.
The first time a 2 appears is at n=105, where the factorization is:
(x-1)*(x^6+x^5+x^4+x^3+x^2+x+1)*(x^4+x^3+x^2+x+1)*
(x^24-x^23+x^19-x^18+x^17-x^16+x^14-x^13+x^12-x^11+x^10-x^8+x^7-x^6+x^5-x+1)*
(x^2+x+1)*(x^12-x^11+x^9-x^8+x^6-x^4+x^3-x+1)*
(x^8-x^7+x^5-x^4+x^3-x+1)*
(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). - N. J. A. Sloane, Apr 18 2008

Mathematica

Table[Max[Abs[Flatten[CoefficientList[Transpose[FactorList[x^i - 1]][[1]], x]]]], {i, 1, 1000}]

Prog

(PARI) a(n) = {my(f = factor(x^n-1)); vecmax(vector(#f~, k, vecmax(apply(x->abs(x), Vec(f[k, 1]))))); } \\ Michel Marcus, Dec 05 2018

Crossrefs

Cf. A013590, A013594.

Sequence in context: A084928 A112316 A112802 * A160338 A216579 A229878

Adjacent sequences: A137976 A137977 A137978 * A137980 A137981 A137982

Keyword

nonn

Author

Ian Miller, Feb 25 2008

Status

approved