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A138475 Least k such that the x^n coefficient of cyclotomic polynomial Phi(k,x) has the largest possible magnitude.

%I A138475 #9 Nov 09 2021 02:49:24
%S A138475 0,1,3,5,5,7,7,105,11,11,11,385,13,429,715,715,165,323323,15015,
%T A138475 323323,1062347,1062347,373065,1062347,11305,1062347,1062347,1062347,
%U A138475 37182145,2800733,37182145,5107219,40755,40755,275147873,10015005,215656441
%N A138475 Least k such that the x^n coefficient of cyclotomic polynomial Phi(k,x) has the largest possible magnitude.
%C A138475 The maximum possible magnitude of the x^n coefficient is A138474(n). Note that a(0)=0 because we assume Phi(0,x)=1; another convention has Phi(0,x)=x, which would force a(0) and a(1) to be reversed.
%C A138475 It appears that (1) for n>80, a(n) has an even number of prime factors and (2) for prime n>80, n divides a(n). Terms up to n=128 were found by exhaustive search; subsequent terms were found by a much faster hill-climbing method.
%D A138475 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 A138475 T. D. Noe, <a href="/A138475/b138475.txt">Table of n, a(n) for n = 0..1000</a>
%H A138475 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 A138475 a(7)=105 because the cyclotomic polynomial Phi(105,x) has the term -2x^7.
%t A138475 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}]; mnN, {k,2,20}]
%Y A138475 Cf. A013594 (smallest order of cyclotomic polynomial containing n or -n as a coefficient).
%Y A138475 Cf. A138474.
%K A138475 nonn
%O A138475 0,3
%A A138475 _T. D. Noe_, Mar 19 2008, Apr 14 2008, Feb 16 2009