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A208547 Phi(k,m) with squarefree k values in sorted order for any integer m > 1.

%I A208547 #11 Jul 08 2015 04:07:39
%S A208547 1,2,6,3,10,5,14,7,15,30,22,11,21,26,42,13,34,17,38,19,33,66,46,23,35,
%T A208547 39,78,70,58,29,62,31,51,102,57,74,114,37,55,82,110,41,86,43,69,138,
%U A208547 94,47,65,210,130,105,106,53,87,174,118,59,77,93,122,186,154
%N A208547 Phi(k,m) with squarefree k values in sorted order for any integer m > 1.
%C A208547 Phi(k,m) denotes cyclotomic polynomial numbers Cyclotomic(k, m).
%C A208547 When k = Product(p_i^j_i), i = 1, 2,..., and p_i are prime factors of k, then Phi(k, m) = Phi(Product(p_i), m^(Product(p_i^(j_i-1)))).
%C A208547 For this reason, number space of Phi(k, m) is still traversed with Phi(k, m) terms with only squarefree k values.
%C A208547 This sequence sorts the Phi(k, m) value along k-axis for all squarefree k values.
%e A208547 For those squarefree numbers that make A000010(k) = 1
%e A208547 Phi(1,m) = -1 + m
%e A208547 Phi(2,m) = 1 + m
%e A208547 Phi(1,m) < Phi(2,m)
%e A208547 So, a(1)=1, a(2)=2;
%e A208547 For those squarefree numbers that make A000010(k) = 2
%e A208547 Phi(3,m) = 1 + m + m^2
%e A208547 Phi(6,m) = 1 - m + m^2
%e A208547 Obviously when integer m > 1, Phi(6,m) < Phi(3,m)
%e A208547 So a(3)=6, a(4)=3 (noting that Phi(6,m) > Phi(2,m) when m > 2, and Phi(6,2) = Phi(2,2))
%e A208547 For those squarefree numbers that make A000010(k) = 4
%e A208547 Phi(5,m) = 1 + m + m^2 + m^3 + m^4
%e A208547 Phi(10,m) = 1 - m + m^2 - m^3 + m^4
%e A208547 Obviously when integer m > 1, Phi(10,m) < Phi(5,m),
%e A208547 So a(5) = 10, and a(6) = 5 (noting Phi(10,m) - Phi(3,m) = m((m^2 + m + 2)(m - 2) + 2) >= 4 > 0 when m >= 2).
%t A208547 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]]; eb = 60; t =  Select[Range[Max[Table[phiinv[n], {n, 1, eb}]]], ((EulerPhi[#] <= eb) && SquareFreeQ[#]) &]; SortBy[t, Cyclotomic[#, 2]&]
%Y A208547 Cf. A005117, A206225.
%K A208547 nonn,easy
%O A208547 1,2
%A A208547 _Lei Zhou_, Feb 28 2012