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

1, 11, 16, 23, 72, 84, 140, 144, 197, 208, 223, 252, 286, 296, 300, 306, 313, 353, 477, 500, 502, 525, 528, 620, 671, 694, 721, 734, 737, 751, 785, 802, 827, 858, 900, 913, 916, 976, 1026, 1056, 1059, 1074, 1080, 1143, 1182, 1197, 1230, 1268, 1281, 1284, 1324

Offset: 1

Author

Amiram Eldar, Mar 14 2024

Keywords

• nonn

Comments

Positions of 0's in A337736.

This sequence has a positive asymptotic density (Shiu, 1991).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• P. Shiu, The distribution of cube-full numbers, Glasgow Mathematical Journal, Vol. 33, No. 3 (1991), pp. 287-295.
• Index entries for sequences related to powerful numbers.

Crossrefs

Cf. A036966, A336175, A337736, A371186.

Programs

Mathematica

cubQ[n_] := (n == 1) || Min @@ FactorInteger[n][[;; , 2]] > 2; Select[Range[1000], ! AnyTrue[Range[#^3 + 1, (# + 1)^3 - 1], cubQ] &]
(* 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]}]; Position[Differences[s], 1] // Flatten]; seq[10^10]

PARI

iscub(n) = n == 1 || vecmin(factor(n)[, 2]) >= 3;
is(n) = for(k = n^3+1, (n+1)^3-1, if(iscub(k), return(0))); 1;

Formula

1 is a term since the two numbers between 1^2 = 1 and (1+1)^2 = 4, 2 and 3, are not cubefull.