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A328253 Nonsquarefree numbers whose first arithmetic derivative (A003415) is not squarefree, but the second derivative (A068346) is.

%I A328253 #8 Oct 11 2019 16:55:49
%S A328253 50,99,125,207,343,375,531,686,725,747,750,819,875,931,1083,1175,1331,
%T A328253 1375,1750,1775,1899,2057,2058,2075,2197,2250,2299,2331,2367,2499,
%U A328253 2525,2625,2750,2853,3250,3425,3430,3577,3610,3771,3789,3843,3875,4059,4149,4250,4311,4394,4459,4475,4626,4693,4750,4775,4875,4913,4998,5145
%N A328253 Nonsquarefree numbers whose first arithmetic derivative (A003415) is not squarefree, but the second derivative (A068346) is.
%H A328253 Antti Karttunen, <a href="/A328253/b328253.txt">Table of n, a(n) for n = 1..10000</a>
%e A328253 50 (= 2 * 5^2) is not squarefree, and its first derivative A003415(50) = 45 = 3^2 * 5 also is not squarefree, but taking derivative yet again, gives A003415(45) = 39 = 3*13, which is squarefree, thus 50 is included in this sequence.
%o A328253 (PARI)
%o A328253 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A328253 isA328253(n) = if(issquarefree(n), 0, my(u=A003415(n)); if(issquarefree(u),0, issquarefree(A003415(u))));
%o A328253 (PARI)
%o A328253 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 A328253 A328248(n) = { my(k=1); while(n && !issquarefree(n), k++; n = A003415checked(n)); (!!n*k); };
%o A328253 isA328253(n) = (3==A328248(n));
%Y A328253 Cf. A003415, A328251.
%Y A328253 Row 4 of array A328250. Indices of 3's in A328248.
%Y A328253 Setwise difference A328245 \ A005117. Intersection of A013929 and A328245.
%K A328253 nonn
%O A328253 1,1
%A A328253 _Antti Karttunen_, Oct 11 2019