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
KEYWORD
nonn
AUTHOR
Jianing Song, May 26 2021
STATUS
approved