Search: id:A138474
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%I A138474
%S A138474 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,
%T A138474 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
%N A138474 Maximum possible magnitude of the x^n coefficient of a cyclotomic polynomial.
%C A138474 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 A138474 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 A138474 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.
%D A138474 H. Moller, Uber die i-ten Koeffizienten der Kreisteilungspolynome, Math. 
               Ann. 188 (1970), 26-38.
%H A138474 T. D. Noe, <a href="b138474.txt">Table of n, a(n) for n=0..1000</a>
%H A138474 Yves Gallot, Pieter Moree and Huib Hommersom, <a href="http://arXiv.org/
               abs/0803.2483">Value distribution of cyclotomic polynomial coefficients</
               a>
%e A138474 a(7)=2 is attained for the cyclotomic polynomial Phi(105,x), which has 
               the term -2x^7.
%t A138474 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 A138474 Cf. A013594 (smallest order of cyclotomic polynomial containing n or 
               -n as a coefficient).
%Y A138474 Sequence in context: A093320 A082370 A005136 this_sequence A058761 A050119 
               A097637
%Y A138474 Adjacent sequences: A138471 A138472 A138473 this_sequence A138475 A138476 
               A138477
%K A138474 nonn
%O A138474 0,8
%A A138474 T. D. Noe (noe(AT)sspectra.com), Mar 19 2008, Apr 14 2008, Feb 16 2009

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