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
Keywords
Examples
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
Links
- N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
Table[Max[Abs[Flatten[CoefficientList[Transpose[FactorList[x^i - 1]][[1]], x]]]], {i, 1, 1000}] -
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
Comments