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).

1, 3, 4, 8, 6, 9, 10, 16, 25, 27, 12, 32, 14, 36, 17, 49, 64, 19, 72, 21, 81, 23, 100, 26, 108, 121, 125, 29, 128, 31, 144, 169, 34, 196, 37, 200, 216, 39, 225, 41, 243, 43, 256, 45, 288, 47, 289, 50, 324, 343, 52, 361, 54, 392, 56, 400, 58, 432, 60, 441, 62, 484

Offset: 1

Author

Michael De Vlieger, Dec 18 2024

Keywords

• nonn
• easy

Comments

The sequence is a list of indices m of powerful numbers a(m).

See comments in A379051 for more information.

Examples

			a(1) = 1 since 1 is powerful, validating the appearance of 1 as an index of a powerful number in the sequence.
a(2) = 3 since self-referential 2 would prove false; 2 is not powerful, but 3 mandates a powerful number a(3).
a(3) = 4 since a(2) = 3, and 4 is the smallest powerful number that has not appeared.
a(4) = 8 since a(3) = 4, and 8 is the smallest powerful number that has not appeared.
a(5) = 6 since m = 5 has not appeared, and 6 is the smallest weak (nonpowerful, in A052485) number k > n.
a(6) = 9 since a(5) = 6, and 9 is the smallest powerful number that has not appeared, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001694, A052485, A121053, A379051, A379053.

Programs

Mathematica

nn = 120;
u = 3; v = {}; w = {}; c = 1;
s = Rest@ Union@ Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[2^30];
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
{1}~Join~Reap[Do[
  If[MemberQ[w, n], k = s[[c]];
    w = DeleteCases[w, n],
    m = Min[{s[[c]], u, v}];
    If[And[Divisible[m, rad[m]^2], CompositeQ[m], n < m],
      AppendTo[v, n]];
      If[Length[v] > 0,
        If[v[[1]] == m, v = Rest[v] ] ]; k = m];
  AppendTo[w, k]; If[k == s[[c]], c++]; Sow[k];
    If[n + 1 >= u, u++;
      While[And[Divisible[u, rad[u]^2], CompositeQ[u]], u++] ], {n, 2, nn}] ][[-1, 1]]