

A138474


Maximum possible magnitude of the x^n coefficient of a cyclotomic polynomial.


3



1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 4, 5, 4, 4, 4, 5, 5, 6, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 9, 9, 7, 8, 8, 10, 13, 12, 10, 12, 9, 11, 15, 13, 13, 14, 15, 13, 16, 15, 15, 14, 16, 24, 17, 21, 21, 16, 22, 28, 26, 23
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OFFSET

0,8


COMMENTS

Terms for n <= 30 come from Table 1 of the Gallot et al. paper, which quotes results from Moller. Sequence A138475 gives the minimum order of the cyclotomic polynomial that produces that maximal coefficient. A very fast method (due to Grytczuk and Tropak) for computing the coefficients up to x^k in the cyclotomic polynomial Phi(n,x) is given by the Mathematica function coef[k,n] below.
The first n for which a(n) > n is 118. The sequence appears to be monotonic for n > 143. Terms up to n=128 were found by exhaustive search; subsequent terms were found by a much faster hillclimbing method.


REFERENCES

A. Grytczuk and B. Tropak, A numerical method for the determination of the cyclotomic polynomial coefficients, Computational number theory (Debrecen, 1989), 1519, de Gruyter, Berlin, 1991.


LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000
Yves Gallot, Pieter Moree and Huib Hommersom, Value distribution of cyclotomic polynomial coefficients, arXiv:0803.2483 [math.NT], 2008.
H. Möller, Über die iten Koeffizienten der Kreisteilungspolynome, Math. Ann. 188 (1970), 2638.


EXAMPLE

a(7)=2 is attained for the cyclotomic polynomial Phi(105,x), which has the term 2x^7.


MATHEMATICA

coef[k_, n_] := Module[{t, b=Table[0, {k+1}]}, t=MoebiusMu[n]*Table[g=GCD[n, km]; MoebiusMu[g]*EulerPhi[g], {m, 0, k1}]; 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^r1}]; mx, {k, 2, 20}]


CROSSREFS

Cf. A013594 (smallest order of cyclotomic polynomial containing n or n as a coefficient).
Sequence in context: A093320 A082370 A005136 * A058761 A050119 A290253
Adjacent sequences: A138471 A138472 A138473 * A138475 A138476 A138477


KEYWORD

nonn


AUTHOR

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


STATUS

approved



