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A371187 Numbers k such that there are no cubefull numbers between k^3 and (k+1)^3.

%I A371187 #8 Mar 14 2024 11:15:14
%S A371187 1,11,16,23,72,84,140,144,197,208,223,252,286,296,300,306,313,353,477,
%T A371187 500,502,525,528,620,671,694,721,734,737,751,785,802,827,858,900,913,
%U A371187 916,976,1026,1056,1059,1074,1080,1143,1182,1197,1230,1268,1281,1284,1324
%N A371187 Numbers k such that there are no cubefull numbers between k^3 and (k+1)^3.
%C A371187 Positions of 0's in A337736.
%C A371187 This sequence has a positive asymptotic density (Shiu, 1991).
%H A371187 Amiram Eldar, <a href="/A371187/b371187.txt">Table of n, a(n) for n = 1..10000</a>
%H A371187 P. Shiu, <a href="https://doi.org/10.1017/S0017089500008351">The distribution of cube-full numbers</a>, Glasgow Mathematical Journal, Vol. 33, No. 3 (1991), pp. 287-295.
%H A371187 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F A371187 1 is a term since the two numbers between 1^2 = 1 and (1+1)^2 = 4, 2 and 3, are not cubefull.
%t A371187 cubQ[n_] := (n == 1) || Min @@ FactorInteger[n][[;; , 2]] > 2; Select[Range[1000], ! AnyTrue[Range[#^3 + 1, (# + 1)^3 - 1], cubQ] &]
%t A371187 (* or *)
%t A371187 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]}]; Position[Differences[s], 1] // Flatten]; seq[10^10]
%o A371187 (PARI) iscub(n) = n == 1 || vecmin(factor(n)[, 2]) >= 3;
%o A371187 is(n) = for(k = n^3+1, (n+1)^3-1, if(iscub(k), return(0))); 1;
%Y A371187 Cf. A036966, A336175, A337736, A371186.
%K A371187 nonn
%O A371187 1,2
%A A371187 _Amiram Eldar_, Mar 14 2024