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A344673 Numbers k such that the expansion of the inverse of the k-th cyclotomic polynomial has a coefficient other than -1, 0 or 1.

%I A344673 #35 Nov 04 2023 13:29:42
%S A344673 561,595,665,741,935,1001,1105,1122,1155,1173,1190,1309,1330,1365,
%T A344673 1463,1479,1482,1495,1615,1683,1729,1767,1785,1870,1955,1995,2001,
%U A344673 2002,2015,2093,2145,2185,2210,2223,2233,2244,2261,2310,2346,2380,2387,2415,2431,2465,2618
%N A344673 Numbers k such that the expansion of the inverse of the k-th cyclotomic polynomial has a coefficient other than -1, 0 or 1.
%C A344673 Define Psi_n(x) = (x^n-1)/Phi_n(x), with Phi_n(x) the n-th cyclotomic polynomial, then 1/Phi_n(x) = -Psi_n(x) * (1 + x^n + x^(2n) + ...), hence the period of coefficients in the expansion of 1/Phi_n(x) is n.
%C A344673 For odd k, k is a term if and only 2*k is a term.
%C A344673 For prime p dividing k, k is a term if and only if p*k is a term.
%C A344673 From _Robert G. Wilson v_, Jun 04 2021: (Start)
%C A344673 The period of the expansion of Psi(n) is A062830(n).
%C A344673 Terms are neither prime nor semiprime.
%C A344673 Least k having a run of j consecutive terms, for j >= 0: 561, 2001, 22630, 68263, ...
%C A344673 (End)
%H A344673 Robert G. Wilson v, <a href="/A344673/b344673.txt">Table of n, a(n) for n = 1..11357</a> (first 900 terms from Jianing Song)
%e A344673 1/Phi_561(x) = 1 - x + x^3 - x^4 + x^6 - x^7 + x^9 - x^10 + x^11 - x^13 + x^14 - x^16 + 2*x^17 + ..., the coefficient of x^17 is 2, so 561 is a term.
%e A344673 1/Phi_595(x) = 1 - x + x^5 - x^6 + x^7 - x^8 + x^10 - x^11 + x^12 - x^13 + x^14 - x^16 + 2*x^17 + ..., the coefficient of x^17 is 2, so 595 is a term.
%t A344673 fQ[n_] := Max@ Union@ Abs@ CoefficientList[ Simplify[(x^n - 1)/Cyclotomic[n, x]], x] > 1; Select[ Range@ 2650, fQ] (* _Robert G. Wilson v_, May 26 2021 *)
%o A344673 (PARI) isA344673(k) = (vecmax(abs(Vec((x^k-1)/polcyclo(k))))>=2)
%Y A344673 Cf. A306453, A333248, A131672, A013590.
%Y A344673 A344706 is a proper subsequence.
%K A344673 nonn
%O A344673 1,1
%A A344673 _Jianing Song_, May 26 2021