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A328303 Numbers whose arithmetic derivative is not squarefree.

%I A328303 #16 Jul 27 2024 19:14:12
%S A328303 0,1,4,8,12,14,15,16,20,24,27,28,32,35,36,39,40,44,46,48,50,51,52,54,
%T A328303 55,56,60,64,65,68,72,76,77,80,81,84,86,87,88,91,92,94,95,96,99,100,
%U A328303 104,108,111,112,115,116,119,120,122,123,124,125,128,132,135,136,138,140,141,143,144,146,148,152,155,156,158,159,160,162,164,168,172,176,180,183,184,187,188,189,192,194,196,200
%N A328303 Numbers whose arithmetic derivative is not squarefree.
%C A328303 Numbers n for which either A003415(n) = 0 or A051903(A003415(n)) > 1.
%H A328303 Antti Karttunen, <a href="/A328303/b328303.txt">Table of n, a(n) for n = 1..10000</a>
%e A328303 Arithmetic derivative of 1 is A003415(1) = 0, which is not a squarefree number (not in A005117), thus 1 is included in this sequence. Ditto for 0, as A003415(0) = 0.
%e A328303 Arithmetic derivative of 8 is A003415(8) = 12 = 2^2 * 3, which is not squarefree, thus 8 is included in this sequence.
%e A328303 Arithmetic derivative of 15 is A003415(15) = 8 = 2^3, which is not squarefree, thus 15 is included in this sequence.
%o A328303 (PARI)
%o A328303 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A328303 isA328303(n) = !issquarefree(A003415(n));
%Y A328303 Cf. A003415, A005117, A008966, A051903.
%Y A328303 Complement of the union of A000040 and A328234.
%Y A328303 Cf. A328245, A328251, A328253, A328304, A328305 (subsequences).
%K A328303 nonn
%O A328303 1,3
%A A328303 _Antti Karttunen_, Oct 13 2019