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Number of factors over Q in the factorization of T_n(x) - 1 where T_n(x) is the Chebyshev polynomial of the first kind.
3

%I #13 Mar 05 2023 07:33:02

%S 1,2,3,4,3,6,3,6,5,6,3,10,3,6,7,8,3,10,3,10,7,6,3,14,5,6,7,10,3,14,3,

%T 10,7,6,7,16,3,6,7,14,3,14,3,10,11,6,3,18,5,10,7,10,3,14,7,14,7,6,3,

%U 22,3,6,11,12,7,14,3,10,7,14,3,22,3,6,11,10,7,14,3,18

%N Number of factors over Q in the factorization of T_n(x) - 1 where T_n(x) is the Chebyshev polynomial of the first kind.

%C If p is an odd prime then a(p) = 3.

%H Antti Karttunen, <a href="/A086369/b086369.txt">Table of n, a(n) for n = 1..2049</a>

%H Y. Z. Gurtas, <a href="https://doi.org/10.4169/amer.math.monthly.124.1.74">Chebyshev polynomials and the minimal polynomial of cos(2pi/n)</a>, Am. Math. Monthly 124 (1) (2017) 73-78, Theorem 1.

%F a(n) = 1+2*A023645(n) for n odd, = 2+2*A023645(n) for n even. [Gurtas] - _R. J. Mathar_, Mar 03 2023

%o (PARI) a(n)={vecsum(factor(polchebyshev(n, 1, x)-1)[, 2])} \\ _Andrew Howroyd_, Jul 10 2018

%Y Cf. A086374.

%K nonn

%O 1,2

%A Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 08 2003

%E a(14) corrected and a(21)-a(80) added by _Andrew Howroyd_, Jul 10 2018