cp's OEIS Frontend

A344706 Odd squarefree 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 A344706 #17 Jun 07 2021 10:16:12
%S A344706 561,595,665,741,935,1001,1105,1155,1173,1309,1365,1463,1479,1495,
%T A344706 1615,1729,1767,1785,1955,1995,2001,2015,2093,2145,2185,2233,2261,
%U A344706 2387,2415,2431,2465,2665,2717,2737,2755,2795,2805,2829,2849,3003,3045,3059,3135,3145,3255
%N A344706 Odd squarefree 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 A344706 Odd squarefree numbers in A344673.
%C A344706 Note that (i) for odd k, Phi_{2*k}(x) = Phi_k(-x); (ii) for prime p dividing k, Phi_{p*k}(x) = Phi_k(x^p). As a result, every term of A344673 can be written as 2^e * (p_1)^(e_1) * (p_2)^(e_2) * ... (p_r)^(e_r) * k, where k is a term of this sequence, p_1, p_2, ..., p_r are the distinct prime factors of k.
%H A344706 Robert G. Wilson v, <a href="/A344706/b344706.txt">Table of n, a(n) for n = 1..1000</a> (first 500 terms from Jianing Song)
%e A344706 665 = 5 * 7 * 19, 1/Phi_665(x) = 1 - x + x^5 - x^6 + x^7 - x^8 + x^10 - x^11 + x^12 - x^13 + x^14 - x^16 + x^17 - x^18 + 2*x^19 + ..., the coefficient of x^19 is 2, so 665 is a term.
%e A344706 1001 = 7 * 11 * 13, 1/Phi_1001(x) = 1 - x + x^7 - x^8 + x^11 - x^12 + x^13 - x^15 + x^18 - x^19 + x^20 - x^23 + x^24 - x^30 + x^31 + x^33 - x^34 + x^35 - x^36 + x^39 - x^41 + x^42 - x^43 + x^44 - x^45 + 2*x^46 + ..., the coefficient of x^46 is 2, so 1001 is a term.
%t A344706 fQ[n_] := Max@ Union@ Abs@ CoefficientList[ Simplify[(x^n - 1)/Cyclotomic[n, x]], x] > 1; Select[1 + 2Range@ 1500, SquareFreeQ@# && fQ@# &] (* _Robert G. Wilson v_, May 29 2021 *)
%o A344706 (PARI) isA344706(k) = (k%2==1) && issquarefree(k) && (vecmax(abs(Vec((x^k-1)/polcyclo(k))))>=2)
%Y A344706 Cf. A306453, A333248, A131672, A154430.
%Y A344706 Proper subsequence of A344673.
%K A344706 nonn
%O A344706 1,1
%A A344706 _Jianing Song_, May 26 2021