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Numbers that are not exponentially cubefree: numbers with at least one noncubefree exponent in their canonical prime factorization.
1

%I #10 Mar 04 2023 08:56:28

%S 256,768,1280,1792,2304,2816,3328,3840,4352,4864,5376,5888,6400,6561,

%T 6912,7424,7936,8448,8960,9472,9984,10496,11008,11520,12032,12544,

%U 13056,13122,13568,14080,14592,15104,15616,16128,16640,17152,17664,18176,18688,19200,19712

%N Numbers that are not exponentially cubefree: numbers with at least one noncubefree exponent in their canonical prime factorization.

%C The asymptotic density of this sequence is 1 - Product_{p prime} ((1 - 1/p) * Sum_{i>=1} 1/p^A004709(i)) = 0.002064412516... .

%H Amiram Eldar, <a href="/A361176/b361176.txt">Table of n, a(n) for n = 1..10000</a>

%e 256 = 2^8 is a term since 8 = 2^3 is not cubefree.

%t noncubfreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] > 2; Select[Range[2*10^4], AnyTrue[FactorInteger[#][[;;, 2]], noncubfreeQ] &]

%o (PARI) isnoncubefree(n) = {n > 7 && vecmax(factor(n)[,2]) > 2; }

%o is(n) = {my(e = factor(n)[, 2]); for(i=1, #e, if(isnoncubefree(e[i]), return(1))); 0; }

%Y Cf. A004709, A046099, A130897, A209061, A262276.

%K nonn

%O 1,1

%A _Amiram Eldar_, Mar 03 2023