A131672 a(n) is the smallest k >= 1 such that the expansion of the inverse of the k-th cyclotomic polynomial has n or -n as a coefficient, or -1 if no such k exists.

1, 561, 1155, 2145, 3795, 5005, 5005, 8645, 8645, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 31395, 31395, 31395, 31395, 31395, 33495, 33495, 33495, 33495, 33495, 33495, 33495, 33495, 33495, 33495, 33495, 33495, 40755

Offset: 1

Author

Jonathan Vos Post, Sep 12 2007

Keywords

• nonn

Comments

Moree's abstract: "Let Psi_n(x) be the monic polynomial having precisely all nonprimitive n-th roots of unity as its simple zeros. One has Psi_n(x) = (x^n-1)/Phi_n(x), with Phi_n(x) the n-th cyclotomic polynomial. The coefficients of Psi_n(x) are integers that like the coefficients of Phi_n(x) tend to be surprisingly small in absolute value, e.g. for n<561 all coefficients of Psi_n(x) are <= 1 in absolute value. We establish various properties of the coefficients of Psi_n(x).

From Jianing Song, May 26 2021: (Start)

The old name was "Arises in reciprocal cyclotomic polynomials".

Since 1/Phi_n(x) = -Psi_n(x) * (1 + x^n + x^(2n) + ...), the period length of coefficients in the expansion of 1/Phi_n(x) is n.

For n > 1, a(n) is odd, composite and squarefree.

Conjecture 1: a(n) > 0 exists for all n.

Conjecture 2: for every k > 2, there exists some positive integer m such that the coefficients in the expansion of 1/Phi_k(x) are -m, -(m-1), ..., m-1, m. If this is true, then for n > 1, a(n) is also the smallest k such that the expansion of 1/Phi_k(x) has both n and -n as coefficients, and a(n) is also the smallest k such that the expansion of 1/Phi_k(x) has a coefficient whose absolute value is greater than or equal to n. (End)

So far, all terms k > 1 have the property that k is squarefree with omega(k) > 2. - Robert G. Wilson v, Jun 09 2021

Examples

			The cyclotomic polynomial Phi_1(x) = 1-x (cf. A013595), so the inverse cyclotomic polynomial Psi_1(x) = 1 (cf. A306453), and so a(1) = 1. - _N. J. A. Sloane_, Jun 08 2021
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..341
• Pieter Moree, Reciprocal cyclotomic polynomials, arXiv:0709.1570 [math.NT], Sep 11 2007, table 1 (computed by Yves Gallot), p. 13.

Crossrefs

Cf. A013595, A306453, A333248, A262404, A262405.

Programs

PARI

a(n) = my(k=1); while((k%2==0) || (isprime(k)) || (!issquarefree(k)) || !setsearch(Set(abs(Vec((x^k-1)/polcyclo(k)))), n), k++); k \\ Jianing Song, May 26 2021

Extensions

New name from Jianing Song, May 26 2021

Terms a(22) onward from Robert G. Wilson v, Jun 09 2021