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A378766 Lexicographically earliest infinite sequence of distinct positive numbers with the property that n is a member of the sequence iff a(n) is powerful (in A001694).

%I A378766 #6 Dec 20 2024 11:37:33
%S A378766 1,3,4,8,6,9,10,16,25,27,12,32,14,36,17,49,64,19,72,21,81,23,100,26,
%T A378766 108,121,125,29,128,31,144,169,34,196,37,200,216,39,225,41,243,43,256,
%U A378766 45,288,47,289,50,324,343,52,361,54,392,56,400,58,432,60,441,62,484
%N A378766 Lexicographically earliest infinite sequence of distinct positive numbers with the property that n is a member of the sequence iff a(n) is powerful (in A001694).
%C A378766 The sequence is a list of indices m of powerful numbers a(m).
%C A378766 See comments in A379051 for more information.
%H A378766 Michael De Vlieger, <a href="/A378766/b378766.txt">Table of n, a(n) for n = 1..10000</a>
%e A378766 a(1) = 1 since 1 is powerful, validating the appearance of 1 as an index of a powerful number in the sequence.
%e A378766 a(2) = 3 since self-referential 2 would prove false; 2 is not powerful, but 3 mandates a powerful number a(3).
%e A378766 a(3) = 4 since a(2) = 3, and 4 is the smallest powerful number that has not appeared.
%e A378766 a(4) = 8 since a(3) = 4, and 8 is the smallest powerful number that has not appeared.
%e A378766 a(5) = 6 since m = 5 has not appeared, and 6 is the smallest weak (nonpowerful, in A052485) number k > n.
%e A378766 a(6) = 9 since a(5) = 6, and 9 is the smallest powerful number that has not appeared, etc.
%t A378766 nn = 120;
%t A378766 u = 3; v = {}; w = {}; c = 1;
%t A378766 s = Rest@ Union@ Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[2^30];
%t A378766 rad[x_] := Times @@ FactorInteger[x][[All, 1]];
%t A378766 {1}~Join~Reap[Do[
%t A378766   If[MemberQ[w, n], k = s[[c]];
%t A378766     w = DeleteCases[w, n],
%t A378766     m = Min[{s[[c]], u, v}];
%t A378766     If[And[Divisible[m, rad[m]^2], CompositeQ[m], n < m],
%t A378766       AppendTo[v, n]];
%t A378766       If[Length[v] > 0,
%t A378766         If[v[[1]] == m, v = Rest[v] ] ]; k = m];
%t A378766   AppendTo[w, k]; If[k == s[[c]], c++]; Sow[k];
%t A378766     If[n + 1 >= u, u++;
%t A378766       While[And[Divisible[u, rad[u]^2], CompositeQ[u]], u++] ], {n, 2, nn}] ][[-1, 1]]
%Y A378766 Cf. A001694, A052485, A121053, A379051, A379053.
%K A378766 nonn,easy
%O A378766 1,2
%A A378766 _Michael De Vlieger_, Dec 18 2024