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

%I A371186 #11 Apr 26 2025 20:55:05
%S A371186 1,2,4,6,8,10,13,15,18,20,23,24,29,32,34,38,39,43,45,48,50,54,57,58,
%T A371186 61,67,69,73,75,77,81,85,88,90,94,96,99,102,105,107,110,113,117,124,
%U A371186 126,128,130,135,137,139,143,147,149,153,158,160,163,167,169,172,176
%N A371186 Indices of the cubes in the sequence of cubefull numbers.
%C A371186 Equivalently, the number of cubefull numbers that do not exceed n^3.
%C A371186 The asymptotic density of this sequence is 1 / A362974 = 0.214626074... .
%C A371186 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.
%H A371186 Amiram Eldar, <a href="/A371186/b371186.txt">Table of n, a(n) for n = 1..10000</a>
%H A371186 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F A371186 A036966(a(n)) = A000578(n) = n^3.
%F A371186 a(n+1) - a(n) = A337736(n) + 1.
%F A371186 a(n) ~ c * n, where c = A362974.
%e A371186 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.
%t A371186 cubQ[n_] := n == 1 || AllTrue[FactorInteger[n], Last[#] >= 3 &]; Position[Select[Range[10^6], cubQ], _?(IntegerQ[Surd[#1, 3]] &)] // Flatten
%t A371186 (* or *)
%t A371186 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]
%o A371186 (PARI) iscub(n) = n == 1 || vecmin(factor(n)[, 2]) >= 3;
%o A371186 lista(kmax) = {my(f, c = 0); for(k = 1, kmax, if(iscub(k), c++; if(ispower(k, 3), print1(c, ", "))));}
%Y A371186 Cf. A000578, A036966, A337736, A362974, A371187.
%Y A371186 Similar sequences: A361936, A371185.
%K A371186 nonn,easy
%O A371186 1,2
%A A371186 _Amiram Eldar_, Mar 14 2024