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).

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, 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, 4, 18, 4, 6, 16, 7, 10, 14, 4, 8, 10, 14, 4, 16, 4, 6, 16, 8, 10, 14, 4, 12, 13

Offset: 0

Author

Wolfdieter Lang, Apr 12 2018

Keywords

• nonn
• easy

Comments

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.

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.

For the number of factors of S(2*n, x) see 2*(tau(2*n+1) - 1) = 2*A095374(n).

Examples

			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).
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).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537

Crossrefs

Cf. A000005, A001227, A001620, A049310, A095374, A187360.

Programs

Mathematica

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 *)

PARI

A001227(n) = numdiv(n>>valuation(n,2));
A302707(n) = (A001227(1+n) + numdiv(2*(n+1)) - 2); \\ Antti Karttunen, Sep 30 2018

Formula

a(n) = tau_{odd}(n+1) + tau(2*(n+1)) - 2, n >= 0, with tau_{odd} = A001227 and tau = A000005.

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).

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

Extensions

Typo in the first formula corrected by Antti Karttunen, Sep 30 2018