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%I #24 Dec 12 2023 10:31:45
%S 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,
%T 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,
%U 3,3,1,2,1,1,3,2,3,3,1,2,1,1,1,4,3,1,3
%N a(n) is the number of real roots of the derivative of the cyclotomic polynomial Phi(n, 1/x).
%F For n = 2^m, a(n) = 0;
%F For odd n = p^m, a(n) = 1;
%F For odd n = p1^r1*p2^r2*...*pm^rm, a(n) = 2m-1;
%F For n = 2*p1^r1*p2^r2*...*pm^rm, a(n) = 2m-1 if p1, ..., pm are odd;
%F For n = 2^r*p1^r1*p2^r2*...*pm^rm, a(n) = 2m if p1, ..., pm are odd and r > 1.
%t c[n_, y_] := Limit[D[Cyclotomic[n, 1/x], x], x -> y]; Table[Length[Solve[c[n, x] == 0, x, Reals]], {n, 1, 128}]
%o (PARI) a(n)=my(v=valuation(n,2)); 2*omega(n>>v) - (v <= 1 && n > 2) \\ _Andrew Howroyd_, Oct 27 2023
%K nonn
%O 1,12
%A _Gevorg Hmayakyan_, Oct 26 2023