|
%I #10 May 13 2023 06:17:22
%S 1,8,16,32,81,128,343,512,729,864,1024,1331,3456,4096,6912,8192,12167,
%T 25000,32768,35937,43904,46656,55296,70304,93312,110592,117649,140608,
%U 186624,287496,331776,357911,373248,592704,707281,889056,1000000,1124864,1157625,1296000
%N Cubefull numbers (A036966) with a record gap to the next cubefull number.
%C This sequence is infinite since the asymptotic density of the cubefull numbers is 0.
%C The corresponding record gaps are 7, 8, 16, 49, 47, 215, 169, 217, 135, 160, ... .
%H Amiram Eldar, <a href="/A363014/b363014.txt">Table of n, a(n) for n = 1..10000</a>
%e 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.
%t 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]
%o (PARI) iscubefull(n) = n==1 || vecmin(factor(n)[, 2]) > 2;
%o 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));}
%Y Cf. A036966, A349062.
%K nonn
%O 1,2
%A _Amiram Eldar_, May 13 2023
|
