A371186 Indices of the cubes in the sequence of cubefull numbers.

1, 2, 4, 6, 8, 10, 13, 15, 18, 20, 23, 24, 29, 32, 34, 38, 39, 43, 45, 48, 50, 54, 57, 58, 61, 67, 69, 73, 75, 77, 81, 85, 88, 90, 94, 96, 99, 102, 105, 107, 110, 113, 117, 124, 126, 128, 130, 135, 137, 139, 143, 147, 149, 153, 158, 160, 163, 167, 169, 172, 176

Offset

1,2

Comments

Equivalently, the number of cubefull numbers that do not exceed n^3.

The asymptotic density of this sequence is 1 / A362974 = 0.214626074... .

If k is a term of A371187 then a(k) and a(k+1) are consecutive integers, i.e., a(k+1) = a(k) + 1.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Index entries for sequences related to powerful numbers.

Formula

A036966(a(n)) = A000578(n) = n^3.

a(n+1) - a(n) = A337736(n) + 1.

a(n) ~ c * n, where c = A362974.

Example

The first 4 cubefull numbers are 1, 8, 16, and 27. The 1st, 2nd, and 4th, 1, 8, and 27, are the first 3 cubes. Therefore, the first 3 terms of this sequence are 1, 2, and 4.

Mathematica

cubQ[n_] := n == 1 || AllTrue[FactorInteger[n], Last[#] >= 3 &]; Position[Select[Range[10^6], cubQ], _?(IntegerQ[Surd[#1, 3]] &)] // Flatten
(* or *)
seq[max_] := Module[{cubs = Union[Flatten[Table[i^5*j^4*k^3, {i, 1, Surd[max, 5]}, {j, 1, Surd[max/i^5, 4]}, {k, Surd[max/(i^5*j^4), 3]}]]], s = {}}, Do[If[IntegerQ[Surd[cubs[[k]], 3]], AppendTo[s, k]], {k, 1, Length[cubs]}]; s]; seq[10^6]

Prog

(PARI) iscub(n) = n == 1 || vecmin(factor(n)[, 2]) >= 3;
lista(kmax) = {my(f, c = 0); for(k = 1, kmax, if(iscub(k), c++; if(ispower(k, 3), print1(c, ", ")))); }

Crossrefs

Cf. A000578, A036966, A337736, A362974, A371187.

Similar sequences: A361936, A371185.

Sequence in context: A186289 A332687 A345981 * A053044 A129011 A130174

Adjacent sequences: A371183 A371184 A371185 * A371187 A371188 A371189

Keyword

nonn,easy

Author

Amiram Eldar, Mar 14 2024

Status

approved