A328244 - OEIS

%I #16 Oct 11 2019 16:56:50

%S 6,9,10,14,18,21,22,25,30,34,38,42,46,50,57,58,62,65,66,69,70,77,78,

%T 82,85,86,93,94,99,105,114,118,121,122,125,126,130,133,134,138,142,

%U 145,146,150,154,161,165,166,169,170,174,177,182,185,186,198,201,202,206,207,209,213,214,217,221,222,230,231,237,238,242,246,253,254,255

%N Numbers whose second arithmetic derivative (A068346) is a squarefree number (A005117).

%C Numbers n for which A008966(A003415(A003415(n))) = 1.

%C Numbers whose first, second or third arithmetic is prime (A157037, A192192, A328239) are all included in this sequence, because: (1) taking arithmetic derivative of a prime gives 1, which is squarefree, (2) primes themselves are squarefree, and (3) only squarefree numbers may have arithmetic derivative that is a prime.

%H Antti Karttunen, <a href="/A328244/b328244.txt">Table of n, a(n) for n = 1..10000</a>

%e For n=6, its first arithmetic derivative is A003415(6) = 5, and its second derivative is A003415(5) = 1, and 1 is a squarefree number (in A005117), thus 6 is included in this sequence.

%e For n=9, A003415(9) = 6, A003415(6) = 5, and 5, like all prime numbers, is squarefree, thus 9 is included in this sequence.

%e For n=14, A003415(14) = 9, A003415(9) = 6 = 2*3, and as 6 is squarefree, 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 isA328244(n) = { my(u=A003415(A003415(n))); (u>0 && issquarefree(u)); };

%Y Cf. A003415, A005117, A008966, A068346, A328234, A328246.

%Y Cf. A157037, A192192, A328239, A328245, A328253 (subsequences).

%K nonn

%O 1,1

%A _Antti Karttunen_, Oct 11 2019