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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
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OFFSET
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1,1
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COMMENTS
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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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LINKS
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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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MATHEMATICA
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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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PROG
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(PARI)
prod_by_bits(bits, fs) = { my(m=1, i=1); while(bits>0, if((bits%2), m *= fs[i]); i++; bits >>= 1); (m); };
A117215(n) = { my(fs=factor('x^n - 1)[, 1], m=0, d, mds=0, k); for(b=0, (2^#fs)-1, d = prod_by_bits(b, fs); k = 0; for(j=0, poldegree(d), k = max(k, abs(polcoeff(d, j)))); if(k==m, mds++, if(k>m, mds=1; m = k))); (mds); }; \\ Antti Karttunen, Jul 01 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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