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%I #22 Mar 29 2024 10:44:39
%S 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,
%T 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,
%U 15,13,16,15,15,14,16,24,17,21,21,16,22,28,26,23
%N Maximum possible magnitude of the x^n coefficient of a cyclotomic polynomial.
%C Terms for n <= 30 come from Table 1 of the Gallot et al. paper, which quotes results from Moller. Sequence A138475 gives the minimum order of the cyclotomic polynomial that produces that maximal coefficient. A very fast method (due to Grytczuk and Tropak) for computing the coefficients up to x^k in the cyclotomic polynomial Phi(n,x) is given by the Mathematica function coef[k,n] below.
%C The first n for which a(n) > n is 118. The sequence appears to be monotonic for n > 143. Terms up to n=128 were found by exhaustive search; subsequent terms were found by a much faster hill-climbing method.
%D 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.
%H T. D. Noe, <a href="/A138474/b138474.txt">Table of n, a(n) for n = 0..1000</a>
%H John Abbott and Nico Mexis, <a href="https://arxiv.org/abs/2403.08751">Cyclotomic Factors and LRS-Degeneracy</a>, arXiv:2403.08751 [math.AC], 2024. See pp. 8-10.
%H Yves Gallot, Pieter Moree and Huib Hommersom, <a href="http://arXiv.org/abs/0803.2483">Value distribution of cyclotomic polynomial coefficients</a>, arXiv:0803.2483 [math.NT], 2008.
%H H. Möller, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002303388">Über die i-ten Koeffizienten der Kreisteilungspolynome</a>, Math. Ann. 188 (1970), 26-38.
%H Carlo Sanna, <a href="https://arxiv.org/abs/2111.04034">A Survey on Coefficients of Cyclotomic Polynomials</a>, arXiv:2111.04034 [math.NT], 2021.
%e a(7)=2 is attained for the cyclotomic polynomial Phi(105,x), which has the term -2x^7.
%t 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}]
%Y Cf. A013594 (smallest order of cyclotomic polynomial containing n or -n as a coefficient).
%K nonn
%O 0,8
%A _T. D. Noe_, Mar 19 2008, Apr 14 2008, Feb 16 2009
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