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A208507 Reordering of A070776 such that the cyclotomic polynomial Phi(A070776, m) is in sorted order for any integer m >= 2.

%I A208507 #15 May 10 2021 02:07:32
%S A208507 1,2,6,4,3,10,12,8,5,14,18,9,7,20,24,16,22,11,26,28,36,13,34,40,48,32,
%T A208507 17,38,54,27,19,44,50,25,46,23,52,56,72,58,29,62,31,68,80,96,64,74,76,
%U A208507 108,37,82,88,100,41,86,98,49,43,92,94,47,104,112,144
%N A208507 Reordering of A070776 such that the cyclotomic polynomial Phi(A070776, m) is in sorted order for any integer m >= 2.
%C A208507 When p is an odd prime number and i >= 1, j >= 1, the cyclotomic polynomial
%C A208507 Phi(2^i*p^j, k)
%C A208507   = Phi(2p,k^(2^(i-1)*p^(j-1)))
%C A208507   = Phi(p, -(k^(2^(i-1)*p^(j-1))))
%C A208507   = (111.....1) (p ones) base -(k^(2^(i-1)*p^(j-1)))
%C A208507 Phi(p^j, k)
%C A208507   = Phi(p, k^(p^(j-1)))
%C A208507   = (111.....1) (p ones) base k^(p^(j-1)).
%C A208507 For odd prime p >= 3, the above numbers can be called "Very Generic Repdigit Numbers".
%C A208507 This sequence is a subsequence of A206225.
%C A208507 The Mathematica program is rewritten to be able to generate this sequence to an arbitrary EulerPhi boundary.
%e A208507 The first 20 elements of A206225 are 1, 2, 6, 4, 3, 10, 12, 8, 5, 14, 18, 9, 7, 15, 20, 24, 16, 30, 22, 11.
%e A208507 Among these, 15 = 3 * 5 and 30 = 2 * 3 * 5 cannot be written in the form 2^i*p^j and are thus rejected. So the first 18 terms of this sequence are 1, 2, 6, 4, 3, 10, 12, 8, 5, 14, 18, 9, 7, 20, 24, 16, 22, 11.
%t A208507 eb = 48; 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]]; elim =  Max[Table[Max[phiinv[n]], {n, 2, eb, 2}]]; t = Select[Range[elim], (a = FactorInteger[#]; b = Length[a]; ((b == 1) || ((b == 2) && (a[[1]][[1]] == 2))) && (EulerPhi[#] <= eb)) &]; SortBy[t, Cyclotomic[#, 2] &]
%Y A208507 Cf. A070776, A206225.
%K A208507 nonn,easy
%O A208507 1,2
%A A208507 _Lei Zhou_, Feb 27 2012