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A086369 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.

%I A086369 #40 Jun 30 2025 04:27:43
%S A086369 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 A086369 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 A086369 22,3,6,11,12,7,14,3,10,7,14,3,22,3,6,11,10,7,14,3,18
%N A086369 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 A086369 If p is an odd prime then a(p) = 3.
%C A086369 a(n) is also the cardinality of the set T containing the divisors d of n and those m > 0 satisfying m + d = n (see the R. J. Mathar formula). Another way of defining a(n) is: if S is the set of nondivisors of n such that r and s belong to S if r + s = n, then a(n) = n - |S|. This second 'co-construction' (since n = |T| + |S|) of a(n) via S is more natural than the direct construction via T, as it avoids two ambiguities in the direct approach. Let f be an involutive function f(x) = y mapping distinct nonzero elements x, y of a set to a pair (x,y) in a set of pairs if x + y = n. Considering T, for m and d in T such that m <> n or d <> n, and m <> d, we have f(m) = d; however, n itself is a member of T yet there exists no valid function f'(n) = 0 since 0 is not a member of T; furthermore, if n is even then there is a unique d in T for which we have to define another function f''(d) = d, valid only for d. Whereas considering S, f(r) = s for every r and s in S and therefore f is a surjective map between S and the set of pairs; then, as stated, n - |S| = |T| = a(n). - _Miles Englezou_, Jun 22 2025
%H A086369 Antti Karttunen, <a href="/A086369/b086369.txt">Table of n, a(n) for n = 1..2049</a>
%H A086369 Yusuf Z. Gürtaş, <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 A086369 a(n) = 1+2*A023645(n) for n odd, = 2+2*A023645(n) for n even. [Gürtaş] - _R. J. Mathar_, Mar 03 2023
%F A086369 a(p^m) = 2*m+1 for prime p > 2 and m >= 1. - _Miles Englezou_, Jun 22 2025
%F A086369 From _Amiram Eldar_, Jun 30 2025: (Start)
%F A086369 a(n) = 2*tau(n) + (n mod 2) - 2, where tau(n) = A000005(n).
%F A086369 Sum_{k=1..n} a(k) ~ 2*n * (log(n) + 2*gamma - 7/4), where gamma is Euler's constant (A001620). (End)
%t A086369 a[n_] := 2 * DivisorSigma[0, n] + Mod[n, 2] - 2; Array[a, 100] (* _Amiram Eldar_, Jun 30 2025 *)
%o A086369 (PARI) a(n)={vecsum(factor(polchebyshev(n, 1, x)-1)[, 2])} \\ _Andrew Howroyd_, Jul 10 2018
%o A086369 (PARI) a(n) = if(n%2==1, 1+2*sumdiv(n, d, d<n/2), 2+2*sumdiv(n, d, d<n/2)) \\ _Miles Englezou_, Jun 22 2025
%o A086369 (PARI) a(n) = 2 * numdiv(n) + n % 2 - 2; \\ _Amiram Eldar_, Jun 30 2025
%Y A086369 Cf. A000005, A001620, A023645, A086374.
%K A086369 nonn
%O A086369 1,2
%A A086369 Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 08 2003
%E A086369 a(14) corrected and a(21)-a(80) added by _Andrew Howroyd_, Jul 10 2018