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.

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

Offset: 1

Author

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

Keywords

• nonn

Comments

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

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

Links

• Antti Karttunen, Table of n, a(n) for n = 1..2049
• Yusuf Z. Gürtaş, Chebyshev polynomials and the minimal polynomial of cos(2pi/n), Am. Math. Monthly 124 (1) (2017) 73-78, Theorem 1.

Crossrefs

Cf. A000005, A001620, A023645, A086374.

Programs

Mathematica

a[n_] := 2 * DivisorSigma[0, n] + Mod[n, 2] - 2; Array[a, 100] (* Amiram Eldar, Jun 30 2025 *)

PARI

a(n)={vecsum(factor(polchebyshev(n, 1, x)-1)[, 2])} \\ Andrew Howroyd, Jul 10 2018

PARI

a(n) = if(n%2==1, 1+2*sumdiv(n, d, dMiles Englezou, Jun 22 2025

PARI

a(n) = 2 * numdiv(n) + n % 2 - 2; \\ Amiram Eldar, Jun 30 2025

Formula

a(n) = 1+2*A023645(n) for n odd, = 2+2*A023645(n) for n even. [Gürtaş] - R. J. Mathar, Mar 03 2023

a(p^m) = 2*m+1 for prime p > 2 and m >= 1. - Miles Englezou, Jun 22 2025

From Amiram Eldar, Jun 30 2025: (Start)

a(n) = 2*tau(n) + (n mod 2) - 2, where tau(n) = A000005(n).

Sum_{k=1..n} a(k) ~ 2*n * (log(n) + 2*gamma - 7/4), where gamma is Euler's constant (A001620). (End)

Extensions

a(14) corrected and a(21)-a(80) added by Andrew Howroyd, Jul 10 2018