A363014 Cubefull numbers (A036966) with a record gap to the next cubefull number.

1, 8, 16, 32, 81, 128, 343, 512, 729, 864, 1024, 1331, 3456, 4096, 6912, 8192, 12167, 25000, 32768, 35937, 43904, 46656, 55296, 70304, 93312, 110592, 117649, 140608, 186624, 287496, 331776, 357911, 373248, 592704, 707281, 889056, 1000000, 1124864, 1157625, 1296000

Offset: 1

Author

Amiram Eldar, May 13 2023

Keywords

• nonn

Comments

This sequence is infinite since the asymptotic density of the cubefull numbers is 0.

The corresponding record gaps are 7, 8, 16, 49, 47, 215, 169, 217, 135, 160, ... .

Examples

			The sequence of cubefull numbers begins with 1, 8, 16, 27, 32, 64, 81 and 125. The differences between these terms are 7, 8, 11, 5, 32, 17 and 44. The record values, 7, 8, 11, 32 and 44 occur after the cubefull numbers 1, 8, 16, 32 and 81, the first 5 terms of this sequence.
		

Links

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

Crossrefs

Cf. A036966, A349062.

Programs

Mathematica

cubQ[n_] := Min[FactorInteger[n][[;; , 2]]] > 2; seq[kmax_] := Module[{s = {}, k1 = 1, gapmax = 0, gap}, Do[If[cubQ[k], gap = k - k1; If[gap > gapmax, gapmax = gap; AppendTo[s, k1]]; k1 = k], {k, 2, kmax}]; s]; seq[10^6]

PARI

iscubefull(n) = n==1 || vecmin(factor(n)[, 2]) > 2;
lista(kmax) = {my(gapmax = 0, gap, k1 = 1); for(k = 2, kmax, if(iscubefull(k), gap = k - k1; if(gap > gapmax, gapmax = gap; print1(k1, ", ")); k1 = k));}