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.

561, 595, 665, 741, 935, 1001, 1105, 1155, 1173, 1309, 1365, 1463, 1479, 1495, 1615, 1729, 1767, 1785, 1955, 1995, 2001, 2015, 2093, 2145, 2185, 2233, 2261, 2387, 2415, 2431, 2465, 2665, 2717, 2737, 2755, 2795, 2805, 2829, 2849, 3003, 3045, 3059, 3135, 3145, 3255

Offset: 1

Author

Jianing Song, May 26 2021

Keywords

• nonn

Comments

Odd squarefree numbers in A344673.

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.

Examples

			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.
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.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1000 (first 500 terms from Jianing Song)

Crossrefs

Cf. A306453, A333248, A131672, A154430.

Proper subsequence of A344673.

Programs

Mathematica

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 *)

PARI

isA344706(k) = (k%2==1) && issquarefree(k) && (vecmax(abs(Vec((x^k-1)/polcyclo(k))))>=2)