A366900 a(n) is the number of real roots of the derivative of the cyclotomic polynomial Phi(n, 1/x).

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

Offset: 1

Author

Gevorg Hmayakyan, Oct 26 2023

Keywords

• nonn

Programs

Mathematica

c[n_, y_] := Limit[D[Cyclotomic[n, 1/x], x], x -> y]; Table[Length[Solve[c[n, x] == 0, x, Reals]], {n, 1, 128}]

PARI

a(n)=my(v=valuation(n,2)); 2*omega(n>>v) - (v <= 1 && n > 2) \\ Andrew Howroyd, Oct 27 2023

Formula

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.