A206944 - OEIS

%I #31 May 10 2021 02:08:31

%S 3,7,11,13,21,31,43,57,61,73,91,111,121,127,133,151,157,183,205,211,

%T 241,273,307,331,341,343,381,421,463,507,521,547,553,601,651,683,703,

%U 757,781,813,871,931,993,1057,1093,1111,1123,1191,1261,1333,1407,1483

%N Numbers Phi_k(m) with integer k > 2, |m| > 1 but k != 2^j (j > 1).

%C Phi_k(m) denotes the cyclotomic polynomial numbers Cyclotomic(k,m).

%C There is a property for Cyclotomic(k,m):

%C Cyclotomic(k^(j+1),m) = Cyclotomic(k,m^(k^j)).

%C So actually when k=2^(j+1), j is a positive integer,

%C Cyclotomic(k,m) = Cyclotomic(2,m^(2^j)) = 1+m^(2^j).

%C If these cases are excluded from A206942, this sequence is obtained.

%C This sequence is a subsequence of A206942.

%C Sequence A059054 is a subsequence of this sequence.

%C The Mathematica program can generate this sequence to arbitrary boundary maxdata without a user's choice of parameters. The parameter determination part of this program is explained at A206864.

%e a(1) = 3 = Phi(6,2).

%e 5 = Phi(4,2) = Phi(2,4) so excluded.

%e a(2) = 7 = Phi(3,2).

%t phiinv[n_, pl_] := Module[{i, p, e, pe, val}, If[pl == {}, Return[If[n == 1, {1}, {}]]]; val = {}; p = Last[pl]; For[e = 0; pe = 1, e == 0 || Mod[n, (p - 1) pe/p] == 0, e++; pe *= p, val = Join[val, pe*phiinv[If[e == 0, n, n*p/pe/(p - 1)], Drop[pl, -1]]]]; Sort[val]]; phiinv[n_] := phiinv[n, Select[1 + Divisors[n], PrimeQ]]; maxdata = 1500; max = Ceiling[(1 + Sqrt[1 + 4*(maxdata - 1)])/4]*2; eb = 2*Floor[(Log[2, maxdata])/2 + 0.5]; While[eg = phiinv[eb]; lu = Length[eg]; lu == 0, eb = eb + 2]; t = Select[Range[eg[[Length[eg]]]], ((EulerPhi[#] <= eb) && ((! IntegerQ[Log[2, #]]) || (# <= 2))) &]; ap = SortBy[t, Cyclotomic[#, 2] &]; an = SortBy[t, Cyclotomic[#, -2] &]; a = {}; Do[i = 2; While[i++; cc = Cyclotomic[ap[[i]], m]; cc <= maxdata,

%t   a = Append[a, cc]]; i = 2; While[i++; cc = Cyclotomic[an[[i]], -m]; cc <= maxdata, a = Append[a, cc]], {m, 2, max}]; Union[a]

%Y Cf. A206942, A194712, A059054, A206864.

%K nonn

%O 1,1

%A _Lei Zhou_, Feb 13 2012