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

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

Author

Robert G. Wilson v, Mar 31 2008

Keywords

• nonn

Comments

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]

Links

• 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.

Crossrefs

Cf. A013594, A046887, A134518.

Programs

Mathematica

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

Extensions

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