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Search: id:A117215
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| A117215 |
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Number of divisors of x^n-1 having the maximal height A114536(n). |
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2
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| 2, 4, 4, 8, 4, 2, 4, 16, 8, 2, 4, 2, 4, 2, 1, 32, 4, 14, 4, 2, 1, 2, 4, 20, 8, 2, 16, 2, 4, 2, 4, 64, 1, 2, 1, 18, 4, 2, 1, 2, 4, 2, 4, 2, 2, 2, 4, 2, 8, 14, 1, 2, 4, 70, 1, 2, 1, 2, 4, 2, 4, 2, 1, 128, 1, 2, 4, 2, 1, 2, 4, 10, 4, 2, 8, 2, 1, 2, 4, 4, 32, 2, 4, 2, 1, 2, 1, 2, 4, 2, 1, 2, 1, 2, 1, 32, 4, 14
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Let p be a prime. Then a(p)=4 because the divisors are x^p-1, x^(p-1)+x^(p-2)+...+1, x-1 and 1. Similarly, a(p^k)=2^(k+1). For n=p*2^k, a(n)=2. For odd primes p and q, a(pq)=1. Conjectures: if n is odd and squarefree, then a(n)=1; if n/2^k is odd and squarefree for k>0, then a(n)=2. All the divisors of x^n-1 are products of cyclotomic polynomials cyclo(d) for various d. When n is the product of distinct odd primes p1..pk, it appears that each cyclotomic index has the form d=p1^e1...pk^ek, where the ei are either 0 or 1 and sum(ei) is odd.
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REFERENCES
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Carl Pomerance and Nathan C. Ryan, "The maximal height of divisors of x^n-1." (To appear in Illinois Journal of Mathematics)
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EXAMPLE
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a(6)=2 because x^3+2x^2+2x+1 and x^3-2x^2+2x-1 both divide x^6-1. In fact, their product is x^6-1.
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MAPLE
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Needs["DiscreteMath`Combinatorica`"]; cyc[n_] := cyc[n] = Cyclotomic[n, x]; PolyHeight[p_] := Max[Abs[CoefficientList[p, x]]]; Table[sd=Subsets[Divisors[n]]; t=Table[PolyHeight[Expand[Product[cyc[sd[[i, j]]], {j, Length[sd[[i]]]}]]], {i, Length[sd]}]; Length[Position[t, Max[t]]], {n, 105}]
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CROSSREFS
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Sequence in context: A070072 A095061 A055077 this_sequence A011173 A162943 A131136
Adjacent sequences: A117212 A117213 A117214 this_sequence A117216 A117217 A117218
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KEYWORD
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nonn
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Mar 03 2006
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