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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.
4
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
1,2
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
LINKS
Pieter Moree, Reciprocal cyclotomic polynomials, arXiv:0709.1570 [math.NT], Sep 11 2007, table 1 (computed by Yves Gallot), p. 13.
EXAMPLE
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
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Vos Post, Sep 12 2007
EXTENSIONS
New name from Jianing Song, May 26 2021
Terms a(22) onward from Robert G. Wilson v, Jun 09 2021
STATUS
approved