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.

561, 595, 665, 741, 935, 1001, 1105, 1122, 1155, 1173, 1190, 1309, 1330, 1365, 1463, 1479, 1482, 1495, 1615, 1683, 1729, 1767, 1785, 1870, 1955, 1995, 2001, 2002, 2015, 2093, 2145, 2185, 2210, 2223, 2233, 2244, 2261, 2310, 2346, 2380, 2387, 2415, 2431, 2465, 2618

Offset

1,1

Comments

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.

For odd k, k is a term if and only 2*k is a term.

For prime p dividing k, k is a term if and only if p*k is a term.

From Robert G. Wilson v, Jun 04 2021: (Start)

The period of the expansion of Psi(n) is A062830(n).

Terms are neither prime nor semiprime.

Least k having a run of j consecutive terms, for j >= 0: 561, 2001, 22630, 68263, ...

(End)

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..11357 (first 900 terms from Jianing Song)

Example

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.
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.

Mathematica

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 *)

Prog

(PARI) isA344673(k) = (vecmax(abs(Vec((x^k-1)/polcyclo(k))))>=2)

Crossrefs

Cf. A306453, A333248, A131672, A013590.

A344706 is a proper subsequence.

Sequence in context: A247906 A278338 A184047 * A344706 A259172 A192297

Adjacent sequences: A344670 A344671 A344672 * A344674 A344675 A344676

Keyword

nonn

Author

Jianing Song, May 26 2021

Status

approved