A013594 -id:A013594 - 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}]

A134518 Least k such that the cyclotomic polynomial Phi(k,x) contains n or -n as a coefficient, where k is restricted to be the product of 3 distinct prime numbers.

Original entry on oeis.org

105, 385, 2431, 2717, 8671, 17119, 19499, 20213, 20213, 34891, 34891, 93439, 93439, 93439, 93439, 93439, 282367, 282367, 282367, 282367, 282367, 282367, 282367, 282367, 282367, 617927, 617927, 849647, 849647, 849647, 849647, 849647, 874507

Offset: 2

T. D. Noe, Oct 29 2007

nonn

It is well-known that k=105 is the first number for which Phi(k,x) has a coefficient other than -1, 0, or 1. See A013594 for the case when k has no restrictions.

Jia Zhao and Xianke Zhang, Coefficients of ternary cyclotomic polynomials, Journal of Number Theory, Volume 130, Issue 10, October 2010, Pages 2223-2237.

Cf. A007304, A013594.

A136418 Smallest order of the cyclotomic polynomial whose maximal coefficient in absolute value is n.

Original entry on oeis.org

0, 105, 385, 1365, 1785, 2805, 3135, 10353, 6545, 12155, 21385, 11165, 21505, 10465, 16555, 19285, 37961, 35105, 18445, 24395, 23205, 53669, 11305, 28595, 17255, 36465, 20615, 42315, 123585, 31535, 49335, 39585, 61295, 35805, 72709, 54285

Offset: 1

Robert G. Wilson v, Mar 31 2008

nonn

This differs from A013594.
For squarefree k, are there an infinite number of cyclotomic polynomials Phi(k,x) of height n? This is true for n=1 because it is known that there are an infinite number of flat cyclotomic polynomials with k the product of three distinct primes. See A117223. - T. D. Noe, Apr 22 2008
There are an infinite number of cyclotomic polynomials of height n if the following generalization of Kaplan's theorem 2 is true: Let N be the product of distinct odd primes and let p be one of those primes. Let q any prime such that q = p (mod N/p), then the height of Phi(Nq/p,x) is the same as the height of Phi(N,x). By Dirichlet's theorem, there are an infinite number of primes q. [From T. D. Noe, Apr 13 2010]

T. D. Noe, Table of n, a(n) for n=1..1000
Nathan Kaplan, Flat cyclotomic polynomials of order three, J. Number Theory 127 (2007), 118-126.

Cf. A013594, A046887, A134518.

f[n_] := f[n] = Max@ Abs@ CoefficientList[ Cyclotomic[n, x], x]; Do[ f@n, {n, 100000}]; t = Array[f, 31000]; Table[ Position[t, n, 1, 1], {n, 25}]//Flatten

More terms from T. D. Noe, Apr 22 2008

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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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A134518 Least k such that the cyclotomic polynomial Phi(k,x) contains n or -n as a coefficient, where k is restricted to be the product of 3 distinct prime numbers.

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A136418 Smallest order of the cyclotomic polynomial whose maximal coefficient in absolute value is n.

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