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A206944 Numbers Phi_k(m) with integer k > 2, |m| > 1 but k != 2^j (j > 1).

%I A206944 #31 May 10 2021 02:08:31
%S A206944 3,7,11,13,21,31,43,57,61,73,91,111,121,127,133,151,157,183,205,211,
%T A206944 241,273,307,331,341,343,381,421,463,507,521,547,553,601,651,683,703,
%U A206944 757,781,813,871,931,993,1057,1093,1111,1123,1191,1261,1333,1407,1483
%N A206944 Numbers Phi_k(m) with integer k > 2, |m| > 1 but k != 2^j (j > 1).
%C A206944 Phi_k(m) denotes the cyclotomic polynomial numbers Cyclotomic(k,m).
%C A206944 There is a property for Cyclotomic(k,m):
%C A206944 Cyclotomic(k^(j+1),m) = Cyclotomic(k,m^(k^j)).
%C A206944 So actually when k=2^(j+1), j is a positive integer,
%C A206944 Cyclotomic(k,m) = Cyclotomic(2,m^(2^j)) = 1+m^(2^j).
%C A206944 If these cases are excluded from A206942, this sequence is obtained.
%C A206944 This sequence is a subsequence of A206942.
%C A206944 Sequence A059054 is a subsequence of this sequence.
%C A206944 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 A206944 a(1) = 3 = Phi(6,2).
%e A206944 5 = Phi(4,2) = Phi(2,4) so excluded.
%e A206944 a(2) = 7 = Phi(3,2).
%t A206944 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 A206944   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 A206944 Cf. A206942, A194712, A059054, A206864.
%K A206944 nonn
%O A206944 1,1
%A A206944 _Lei Zhou_, Feb 13 2012