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

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

Offset: 0

Author

T. D. Noe, Mar 19 2008, Apr 14 2008, Feb 16 2009

Keywords

• nonn

Comments

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.

Examples

			a(7)=105 because the cyclotomic polynomial Phi(105,x) 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
• 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).

Cf. A138474.

Programs

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}]; mnN, {k,2,20}]