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Sylvester cyclotomic numbers for A002605.
2

%I #11 Feb 16 2025 08:32:57

%S 1,2,6,8,44,10,328,56,408,76,18272,52,136384,568,3856,3104,7598336,

%T 424,56714752,2896,215104,31648,3159738368,3088,536013824,236224,

%U 71910912,161344,1313964867584,2320,9807567290368,9634304,667730944,13160704,37860806656,172864,4078438577864704,98232832,37201186816,9584896

%N Sylvester cyclotomic numbers for A002605.

%C Primitive parts of A002605.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SylvesterCyclotomicNumber.html">Sylvester Cyclotomic Number</a>.

%F a(n) = A002605(n)/A105606(n); a(n) = product{k=1..n-1, if(gcd(n, k)=1, (1+sqrt(3))-(1-sqrt(3))*exp(2*Pi*I*k/n), 1)}, I=sqrt(-1). - _Robert G. Wilson v_, Aug 02 2005

%t f[n_] := FullSimplify[ Expand[ Times @@ ((1+Sqrt[3])-(1-Sqrt[3])*Exp[2Pi*I*Select[Range[n], GCD[ #, n] == 1 &]/n])]]; Table[ f[n], {n, 2, 32] (* _Robert G. Wilson v_, Aug 02 2005 *)

%Y Cf. A002605, A105608, A105609.

%K nonn,changed

%O 1,2

%A _Paul Barry_, Apr 15 2005

%E More terms from _David Wasserman_, May 06 2008