A086374 -id:A086374 - cp's OEIS frontend

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

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

nonn

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

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.

Cf. A000005, A001620, A023645, A086374.

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

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

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

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

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)

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

A086375 Number of factors over Q in the factorization of U_n(x) + 1 where U_n(x) is the Chebyshev polynomial of the second kind.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 3, 3, 6, 2, 4, 4, 5, 4, 5, 2, 7, 4, 4, 4, 8, 3, 4, 5, 6, 4, 8, 2, 8, 4, 3, 6, 9, 4, 5, 4, 8, 4, 8, 2, 8, 6, 4, 6, 10, 3, 6, 5, 7, 4, 8, 4, 10, 6, 4, 4, 12, 2, 6, 6, 7, 8, 7, 4, 8, 4, 8, 4, 14, 2, 5, 6, 6, 8, 8, 4, 12, 5, 4, 5, 12, 4, 6, 6, 8, 4, 12, 4, 10, 6, 4, 6, 12, 4, 6, 6, 10, 6, 9

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 06 2003

			a(7)=3 because 1+U(7,x)=1+128x^7-192x^5+80x^3-8x=(2x+1)(8x^3-6x+1)(8x^3-4x^2-4x+1).

Cf. A086327.

Cf. A086374.

p2 = 1; p1 = 2*x; for (n = 1, 103, p = 2*x*p1 - p2; f = factor(p1 + 1); print(sum(i = 1, matsize(f)[1], f[i, 2]), " "); p2 = p1; p1 = p); \\ David Wasserman, Mar 02 2005

More terms from David Wasserman and Emeric Deutsch, Mar 02 2005

A129572 A129372 * A097806.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Gary W. Adamson, Apr 22 2007

Row sums = A086374: (1, 2, 3, 2, 3, 4, 3, 2, 5, 4, ...). A129573 = A097806 * A129372.

			First few rows of the triangle:
  1;
  1, 1;
  1, 1, 1;
  0, 0, 1, 1;
  1, 0, 0, 1, 1;
  1, 1, 0, 0, 1, 1;
  1, 0, 0, 0, 0, 1, 1;
  0, 0, 0, 0, 0, 0, 1, 1;
  ...

Cf. A129372, A097806, A086374, A129573.

A129372 * A097806 as infinite lower triangular matrices.

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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.

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A086375 Number of factors over Q in the factorization of U_n(x) + 1 where U_n(x) is the Chebyshev polynomial of the second kind.

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PARI

Extensions

A129572 A129372 * A097806.

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