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A138475
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Least k such that the x^n coefficient of cyclotomic polynomial Phi(k,x) has the largest possible magnitude.
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3
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0, 1, 3, 5, 5, 7, 7, 105, 11, 11, 11, 385, 13, 429, 715, 715, 165, 323323, 15015, 323323, 1062347, 1062347, 373065, 1062347, 11305, 1062347, 1062347, 1062347, 37182145, 2800733, 37182145, 5107219, 40755, 40755, 275147873, 10015005, 215656441
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OFFSET
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0,3
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COMMENTS
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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.
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.
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REFERENCES
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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.
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LINKS
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EXAMPLE
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a(7)=105 because the cyclotomic polynomial Phi(105,x) has the term -2x^7.
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A013594 (smallest order of cyclotomic polynomial containing n or -n as a coefficient).
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KEYWORD
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nonn
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AUTHOR
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T. D. Noe, Mar 19 2008, Apr 14 2008, Feb 16 2009
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STATUS
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approved
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