A138474 -id:A138474 - cp's OEIS frontend

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

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

nonn

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.

			a(7)=105 because the cyclotomic polynomial Phi(105,x) has the term -2x^7.

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.

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.

Cf. A013594 (smallest order of cyclotomic polynomial containing n or -n as a coefficient).

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

A140671 Numbers useful in computing A(k), the largest possible magnitude of the x^k coefficient in a cyclotomic polynomial.

Original entry on oeis.org

150, 820, 2751, 9298, 18016, 38907, 62286, 110681

Offset: 3

T. D. Noe, May 21 2008

nonn

The values of A(k) are in sequence A138474. The least order N(k) of the cyclotomic polynomial producing A(k) is A138475(k). For k >= a(n), computational experiments have shown that only primes >= prime(n) divide N(k). These terms were determined empirically; is there another way of characterizing these numbers?

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

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A140671 Numbers useful in computing A(k), the largest possible magnitude of the x^k coefficient in a cyclotomic polynomial.

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