A206292 - OEIS

%I #30 Jul 14 2021 10:03:07

%S 1,2,3,4,6,12,18,30,42,48,60,66,70,78,90,102,120,126,150,180,210,240,

%T 270,300,330,420,450,462,480,510,540,630,660,690,780,840,870,924,1050,

%U 1092,1140,1260,1320,1470,1560,1680,1890,2310,2730,2940,3150,3570,3990

%N Numbers k such that cyclotomic polynomial Phi(k,-m) < Phi(j,-m) for any j > k and m >= 2.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cyclotomic_polynomial">Cyclotomic polynomial</a>

%e For k such that A000010(k) = 1:

%e   Phi(1, -m) = -1 - m,

%e   Phi(2, -m) = 1 - m,

%e   Phi(1, -m) <  Phi(2, -m),

%e   so a(1) = 1, a(2) = 2.

%e For k > 2 such that A000010(k) = 2:

%e   Phi(3, -m) = 1 - m + m^2,

%e   Phi(4, -m) = 1 + m^2,

%e   Phi(6, -m) = 1 + m + m^2.

%e When integer m > 1, Phi(3, -m) < Phi(4, -m) < Phi(6, -m), so a(3) = 3, a(4) = 4, and a(5) = 6.

%e For k > 6 such that A000010(k) = 4:

%e   Phi(8, -m) = 1 + m^4,

%e   Phi(10, -m) = 1 + m + m^2 + m^3 + m^4,

%e   Phi(12, -m) = 1 - m^2 + m^4.

%e When integer m > 1, Phi(12, -m) < Phi(8, -m) < Phi(10, -m), so a(6) = 12.

%t t = Select[Range[4000], EulerPhi[#] <= 1000 &]; t =  SortBy[t, Cyclotomic[#, -2] &]; DeleteDuplicates[Table[Max[Take[t, n]], {n, 1, Length[t]}]]

%Y Cf. A194712, A206225, A000010, A002202, A032447.

%K nonn

%O 1,2

%A _Lei Zhou_, Feb 13 2012