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

1,2

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, ... .

Links

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

Example

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.

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]

Prog

(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)); }

Crossrefs

Cf. A036966, A349062.

Sequence in context: A335850 A206375 A146541 * A261976 A291124 A239280

Adjacent sequences: A363011 A363012 A363013 * A363015 A363016 A363017

Keyword

nonn

Author

Amiram Eldar, May 13 2023

Status

approved