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A136418 Smallest order of the cyclotomic polynomial whose maximal coefficient in absolute value is n. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

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

CROSSREFS

Cf. A013594, A046887, A134518.

Sequence in context: A102792 A013594 A160340 * A134518 A160717 A165056

Adjacent sequences:  A136415 A136416 A136417 * A136419 A136420 A136421

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Mar 31 2008

EXTENSIONS

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

STATUS

approved

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Last modified April 1 05:04 EDT 2020. Contains 333155 sequences. (Running on oeis4.)