A138474 Maximum possible magnitude of the x^n coefficient of a cyclotomic polynomial.
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 4, 5, 4, 4, 4, 5, 5, 6, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 9, 9, 7, 8, 8, 10, 13, 12, 10, 12, 9, 11, 15, 13, 13, 14, 15, 13, 16, 15, 15, 14, 16, 24, 17, 21, 21, 16, 22, 28, 26, 23
Offset: 0
Keywords
Examples
a(7)=2 is attained for the cyclotomic polynomial Phi(105,x), which has the term -2x^7.
References
- A. Grytczuk and B. Tropak, A numerical method for the determination of the cyclotomic polynomial coefficients, Computational number theory (Debrecen, 1989), 15-19, de Gruyter, Berlin, 1991.
Links
- T. D. Noe, Table of n, a(n) for n = 0..1000
- John Abbott and Nico Mexis, Cyclotomic Factors and LRS-Degeneracy, arXiv:2403.08751 [math.AC], 2024. See pp. 8-10.
- Yves Gallot, Pieter Moree and Huib Hommersom, Value distribution of cyclotomic polynomial coefficients, arXiv:0803.2483 [math.NT], 2008.
- H. Möller, Über die i-ten Koeffizienten der Kreisteilungspolynome, Math. Ann. 188 (1970), 26-38.
- Carlo Sanna, A Survey on Coefficients of Cyclotomic Polynomials, arXiv:2111.04034 [math.NT], 2021.
Crossrefs
Cf. A013594 (smallest order of cyclotomic polynomial containing n or -n as a coefficient).
Programs
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Mathematica
coef[k_,n_] := Module[{t, b=Table[0,{k+1}]}, t=-MoebiusMu[n]*Table[g=GCD[n,k-m]; MoebiusMu[g]*EulerPhi[g], {m,0,k-1}]; b[[1]]=1; Do[b[[j+1]] = Take[b,j].Take[t,-j]/j, {j,k}]; b]; Table[mx=1; r=PrimePi[k]+1; mnN=Prime[r]; ps=Reverse[Prime[Range[r]]]; Do[d=IntegerDigits[i,2,r]; n=Times@@Pick[ps,d,1]; c=Abs[coef[k,n][[ -1]]]; If[c==mx, mnN=Min[mnN,n], If[c>mx, mx=c; mnN=n]], {i,2^r-1}]; mx, {k,2,20}]
Comments