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A366900
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a(n) is the number of real roots of the derivative of the cyclotomic polynomial Phi(n, 1/x).
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0
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0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 3, 0, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 0, 3, 1, 3, 2, 1, 1, 3, 2, 1, 3, 1, 2, 3, 1, 1, 2, 1, 1, 3, 2, 1, 1, 3, 2, 3, 1, 1, 4, 1, 1, 3, 0, 3, 3, 1, 2, 3, 3, 1, 2, 1, 1, 3, 2, 3, 3, 1, 2, 1, 1, 1, 4, 3, 1, 3
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OFFSET
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1,12
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LINKS
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FORMULA
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For n = 2^m, a(n) = 0;
For odd n = p^m, a(n) = 1;
For odd n = p1^r1*p2^r2*...*pm^rm, a(n) = 2m-1;
For n = 2*p1^r1*p2^r2*...*pm^rm, a(n) = 2m-1 if p1, ..., pm are odd;
For n = 2^r*p1^r1*p2^r2*...*pm^rm, a(n) = 2m if p1, ..., pm are odd and r > 1.
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MATHEMATICA
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c[n_, y_] := Limit[D[Cyclotomic[n, 1/x], x], x -> y]; Table[Length[Solve[c[n, x] == 0, x, Reals]], {n, 1, 128}]
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PROG
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(PARI) a(n)=my(v=valuation(n, 2)); 2*omega(n>>v) - (v <= 1 && n > 2) \\ Andrew Howroyd, Oct 27 2023
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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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