cp's OEIS Frontend

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.

%I A131672 #36 Jan 25 2024 07:54:00
%S A131672 1,561,1155,2145,3795,5005,5005,8645,8645,11305,11305,11305,11305,
%T A131672 11305,11305,11305,11305,11305,11305,11305,11305,31395,31395,31395,
%U A131672 31395,31395,33495,33495,33495,33495,33495,33495,33495,33495,33495,33495,33495,33495,40755
%N 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.
%C A131672 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).
%C A131672 From _Jianing Song_, May 26 2021: (Start)
%C A131672 The old name was "Arises in reciprocal cyclotomic polynomials".
%C A131672 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.
%C A131672 For n > 1, a(n) is odd, composite and squarefree.
%C A131672 Conjecture 1: a(n) > 0 exists for all n.
%C A131672 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)
%C A131672 So far, all terms k > 1 have the property that k is squarefree with omega(k) > 2. - _Robert G. Wilson v_, Jun 09 2021
%H A131672 Robert G. Wilson v, <a href="/A131672/b131672.txt">Table of n, a(n) for n = 1..341</a>
%H A131672 Pieter Moree, <a href="http://arXiv.org/abs/0709.1570">Reciprocal cyclotomic polynomials</a>, arXiv:0709.1570 [math.NT], Sep 11 2007, table 1 (computed by Yves Gallot), p. 13.
%e A131672 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
%o A131672 (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
%Y A131672 Cf. A013595, A306453, A333248, A262404, A262405.
%K A131672 nonn
%O A131672 1,2
%A A131672 _Jonathan Vos Post_, Sep 12 2007
%E A131672 New name from _Jianing Song_, May 26 2021
%E A131672 Terms a(22) onward from _Robert G. Wilson v_, Jun 09 2021