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A302707 Number of factors of Chebyshev polynomial S(2*n+1, x) (A049310) over the integers. Factorization is into the minimal integer polynomials C (A187360).

%I A302707 #18 Feb 03 2025 02:12:46
%S A302707 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 A302707 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 A302707 4,18,4,6,16,7,10,14,4,8,10,14,4,16,4,6,16,8,10,14,4,12,13
%N A302707 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 A302707 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 A302707 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 A302707 For the number of factors of S(2*n, x) see 2*(tau(2*n+1) - 1) = 2*A095374(n).
%H A302707 Antti Karttunen, <a href="/A302707/b302707.txt">Table of n, a(n) for n = 0..65537</a>
%F A302707 a(n) = tau_{odd}(n+1) + tau(2*(n+1)) - 2, n >= 0, with tau_{odd} = A001227 and tau = A000005.
%F A302707 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).
%F A302707 Sum_{k=1..n} a(k) ~ 2*n * (log(n) + 2*gamma + log(2)/2 - 2), where gamma is Euler's constant (A001620). - _Amiram Eldar_, Feb 03 2025
%e A302707 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 A302707 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).
%t A302707 a[n_] := DivisorSigma[0, 2*(n+1)] + DivisorSigma[0, (n+1)/2^IntegerExponent[n+1, 2]] - 2; Array[a, 100, 0] (* _Amiram Eldar_, Feb 03 2025 *)
%o A302707 (PARI)
%o A302707 A001227(n) = numdiv(n>>valuation(n,2));
%o A302707 A302707(n) = (A001227(1+n) + numdiv(2*(n+1)) - 2); \\ _Antti Karttunen_, Sep 30 2018
%Y A302707 Cf. A000005, A001227, A001620, A049310, A095374, A187360.
%K A302707 nonn,easy
%O A302707 0,2
%A A302707 _Wolfdieter Lang_, Apr 12 2018
%E A302707 Typo in the first formula corrected by _Antti Karttunen_, Sep 30 2018