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%I #13 Mar 06 2021 01:32:26
%S 1,2,4,3,4,6,4,4,7,6,4,8,4,6,10,5,4,10,4,8,10,6,4,10,7,6,10,8,4,14,4,
%T 6,10,6,10,13,4,6,10,10,4,14,4,8,16,6,4,12,7,10,10,8,4,14,10,10,10,6,
%U 4,18,4,6,16,7,10,14,4,8,10,14,4,16,4,6,16,8,10,14,4,12,13
%N Number of factors of Chebyshev polynomial S(2*n+1, x) (A049310) over the integers. Factorization is into the minimal integer polynomials C (A187360).
%C For the factorization of the Chebyshev S polynomials (coefficients in A049310) with odd index into the minimal polynomials of {2*cos(Pi/k)}_{k>=1} (coefficients in A187360) see an Apr 12 2018 comment in A049310.
%C Note that factors -C(k, -x) may appear also and they come always together with C(k, x) (the minus signs are not counted as factors here). C(2, x) = x is always a factor.
%C For the number of factors of S(2*n, x) see 2*(tau(2*n+1) - 1) = 2*A095374(n).
%H Antti Karttunen, <a href="/A302707/b302707.txt">Table of n, a(n) for n = 0..65537</a>
%F a(n) = tau_{odd}(n+1) + tau(2*(n+1)) - 2, n >= 0, with tau_{odd} = A001227 and tau = A000005.
%F G.f.: Sum_{k>=1} (x^(k-1)/(1-x^(2*k)) + x^(k-1)*(2+x^k)/(1-x^(2*k))) - 2/(1-x).
%e a(2) = 4 because S(5, x) = 3*x-4*x^3+x^5 = x*(-1 + x)*(1 + x)*(-3 + x^2) = C(2, x)*C(3, x)*(-C(3, -x))*C(6, x).
%e a(5) = 6 because S(11, x) = -6*x + 35*x^3 - 56*x^5 + 36*x^7 - 10*x^9 + x^11 = x*(-1 + x)*(1 + x)*(-2 + x^2)*(-3 + x^2)*(1 - 4*x^2 + x^4) = C(2, x)*C(3, x)*(-C(3, -x))*C(4, x)*C(6, x)*C(12, x).
%o (PARI)
%o A001227(n) = numdiv(n>>valuation(n,2));
%o A302707(n) = (A001227(1+n) + numdiv(2*(n+1)) - 2); \\ _Antti Karttunen_, Sep 30 2018
%Y Cf. A000005, A001227, A049310, A095374, A187360.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Apr 12 2018
%E Typo in the first formula corrected by _Antti Karttunen_, Sep 30 2018
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