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

1,2

Comments

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

See comments in A379051 for more information.

Links

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

Example

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.

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]]

Crossrefs

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

Sequence in context: A213954 A357067 A074212 * A125715 A129283 A332844

Adjacent sequences: A378763 A378764 A378765 * A378767 A378768 A378769

Keyword

nonn,easy

Author

Michael De Vlieger, Dec 18 2024

Status

approved