login
Numbers whose second arithmetic derivative (A068346) is a squarefree number (A005117), but the first derivative (A003415) is not.
7

%I #11 Oct 11 2019 16:56:39

%S 14,46,50,65,77,86,94,99,122,125,138,146,207,230,302,334,343,346,375,

%T 426,531,546,554,581,590,626,662,682,686,710,717,718,725,734,747,750,

%U 819,842,869,875,931,965,1002,1041,1083,1130,1145,1146,1166,1175,1202,1241,1265,1310,1331,1337,1349,1375,1418,1461,1466,1469,1501,1529,1541

%N Numbers whose second arithmetic derivative (A068346) is a squarefree number (A005117), but the first derivative (A003415) is not.

%H Antti Karttunen, <a href="/A328245/b328245.txt">Table of n, a(n) for n = 1..10000</a>

%e For n = 14, its first arithmetic derivative, A003415(14) = 9 = 3^2 is not squarefree, while the second arithmetic derivative, A003415(9) = 6 = 2* 3 is, thus 14 is included in this sequence.

%o (PARI)

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o isA328245(n) = { my(u=A003415(n)); (!issquarefree(u) && issquarefree(A003415(u))); }; \\ issquarefree(0) returns 0 as zero is not considered as a squarefree number.

%Y Cf. A003415, A005117, A008966, A068346.

%Y Setwise difference A328244 \ A328234.

%Y Cf. A328253 (a subsequence, nonsquarefree terms).

%K nonn

%O 1,1

%A _Antti Karttunen_, Oct 11 2019