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A328305 Numbers that are cubefree, but not squarefree and whose first arithmetic derivative is not squarefree, but some k-th (with k >= 2) derivative is.

%I A328305 #10 Oct 13 2019 18:09:45
%S A328305 50,99,207,306,531,549,725,747,819,931,1083,1175,1611,1775,1899,2057,
%T A328305 2075,2299,2331,2367,2499,2525,2842,2853,2891,3425,3577,3610,3771,
%U A328305 3789,3843,4059,4149,4311,4475,4575,4626,4693,4775,4998,5239,5274,5341,5547,5634,5706,5715,5746,5819,5949,6147,6223,6275,6381,6413,6475,6575
%N A328305 Numbers that are cubefree, but not squarefree and whose first arithmetic derivative is not squarefree, but some k-th (with k >= 2) derivative is.
%C A328305 Numbers n for which A051903(n) = 2 and A328248(n) > 2.
%H A328305 Antti Karttunen, <a href="/A328305/b328305.txt">Table of n, a(n) for n = 1..10000</a>
%e A328305 50 is not squarefree, as 50 = 2 * 5^2, and neither its arithmetic derivative A003415(50) = 45 = 3^2 * 5 is squarefree, but its second derivative A003415(45) = 39 = 3*13 is, thus 50 is included in this sequence.
%o A328305 (PARI)
%o A328305 A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));
%o A328305 A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
%o A328305 A328248(n) = { my(k=1); while(n && !issquarefree(n), k++; n = A003415checked(n)); (!!n*k); };
%o A328305 isA067259(n) = (2==A051903(n));
%o A328305 isA328305(n) = (isA067259(n)&&(A328248(n)>2));
%Y A328305 Cf A003415, A051903, A328248, A328253.
%Y A328305 Subsequence of A067259, A328303, A328304 and A328321.
%K A328305 nonn
%O A328305 1,1
%A A328305 _Antti Karttunen_, Oct 13 2019