This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A105609 #30 Feb 16 2025 08:32:57
%S A105609 1,0,-1,-2,1,-3,-1,2,-1,5,-1,1,1,-7,1,2,1,-3,-1,1,1,-11,-1,1,1,13,-1,
%T A105609 1,1,1,-1,2,1,17,1,1,1,-19,1,1,1,1,-1,1,1,-23,-1,1,-1,5,1,1,1,-3,1,1,
%U A105609 1,29,-1,1,1,-31,1,2,1,1,-1,1,1,1,-1,1,1,37
%N A105609 Sylvester numbers for 1/(1+x^2).
%H A105609 Nathaniel Johnston, <a href="/A105609/b105609.txt">Table of n, a(n) for n = 0..2500</a>
%H A105609 Peter Luschny and Stefan Wehmeier, <a href="http://arxiv.org/abs/0909.1838">The lcm(1,2,...,n) as a product of sine values sampled over the points in Farey sequences</a>, arXiv:0909.1838 [math.CA], 2009.
%H A105609 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SylvesterCyclotomicNumber.html">Sylvester Cyclotomic Number</a>.
%F A105609 a(n) = Product_{k=1..n} if(gcd(n+1, k)=1, (I+I*exp(2*Pi*I*k/(n+1))), 1), I=sqrt(-1).
%F A105609 alpha(n) = Product_{0<k<n} if(gcd(k,n)=1, 2*cos(Pi*k/n), 1); a(n) = alpha(n+1). Cf. the formula in A014963 with cos replaced by sin. - _Peter Luschny_, Jun 09 2011
%e A105609 (x+I)(x-I)=1+x^2
%p A105609 A105609 := proc(n)local k: round(Re(mul(evalf(`if`(gcd(n+1, k)=1, I+I*exp(2*Pi*I*k/(n+1)), 1)),k=1..n))): end:
%p A105609 seq(A105609(n),n=0..20); # _Nathaniel Johnston_, Apr 20 2011
%p A105609 A105609 := proc(n) local k; mul(`if`(igcd(n+1,k)=1, 2*cos(Pi*k/(n+1)), 1), k=1..n) end; seq(round(A105609(n)), n = 0..73); # _Peter Luschny_, Jun 09 2011
%t A105609 f[n_] := FullSimplify[ Expand[Times @@ (I + I*Exp[2Pi*I*Select[Range[n], GCD[ #, n] == 1 &]/n])]]; Table[ f[n], {n, 0, 32}] (* _Robert G. Wilson v_, Aug 02 2005 *)
%Y A105609 Cf. A020513, A105608, A105607.
%K A105609 easy,sign
%O A105609 0,4
%A A105609 _Paul Barry_, Apr 15 2005
%E A105609 a(40)-a(73) from _Nathaniel Johnston_, Apr 20 2011