cp's OEIS Frontend

A328234 Numbers whose arithmetic derivative (A003415) is a squarefree number (A005117) > 1.

%I A328234 #30 Dec 01 2024 16:28:00
%S A328234 6,9,10,18,21,22,25,26,30,33,34,38,42,45,49,57,58,62,63,66,69,70,74,
%T A328234 75,78,82,85,90,93,98,102,105,106,110,114,117,118,121,126,129,130,133,
%U A328234 134,142,145,147,150,153,154,161,165,166,169,170,171,174,175,177,178,182,185,186,190,195,198,201,202,205,206,209,210,213
%N A328234 Numbers whose arithmetic derivative (A003415) is a squarefree number (A005117) > 1.
%C A328234 Sequence A328393 without primes.
%C A328234 No multiples of 4 because this is a subsequence of A048103.
%C A328234 All terms are cubefree, but being a cubefree non-multiple of 4 doesn't guarantee a membership, as for example 99 = 3^2 * 11 has an arithmetic derivative 11*(2*3) + 3^2 = 75 = 5^2 * 3, and thus is not included in this sequence. (See e.g., A328305).
%H A328234 Antti Karttunen, <a href="/A328234/b328234.txt">Table of n, a(n) for n = 1..10000</a>
%t A328234 arthD[n_]:=Module[{fi=FactorInteger[n]},n Total[(fi[[;;,2]]/fi[[;;,1]])]]; Select[Range[300],arthD[#]>1&&SquareFreeQ[arthD[#]]&] (* _Harvey P. Dale_, Dec 01 2024 *)
%o A328234 (PARI)
%o A328234 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A328234 isA328234(n) = { my(u=A003415(n)); (u>1 && issquarefree(u)); };
%Y A328234 Cf. A003415, A005117, A068328, A068719, A235991, A327862, A328239, A328242, A328244, A328245, A328246, A328247.
%Y A328234 Cf. A328252 (nonsquarefree terms), A157037, A192192, A327978 (other subsequences).
%Y A328234 Subsequence of following sequences: A004709, A048103, A328393.
%Y A328234 Complement of the union of A000040 and A328303, i.e., complement of A328303, but without primes.
%Y A328234 Cf. also A328248, A328250, A328305.
%K A328234 nonn
%O A328234 1,1
%A A328234 _Antti Karttunen_, Oct 10 2019