A342009 - OEIS

%I #13 Mar 12 2021 23:10:39

%S 1,2,4,8,9,12,15,16,20,32,36,48,64,81,108,112,128,144,180,189,192,196,

%T 225,256,320,324,400,432,500,512,528,576,625,729,768,972,1024,1225,

%U 1296,1300,1360,1452,1728,2048,2160,2304,2700,2816,2916,3024,3072,3375,3564,3840,3888,3993,4096,4800,5120,5184,5292,5616,6000

%N Numbers k such that the arithmetic derivative of k is a multiple of phi(k).

%C Numbers k for which A000010(k) is a divisor of A003415(k), or equally, k for which A173557(k) is a divisor of A342001(k).

%H Antti Karttunen, <a href="/A342009/b342009.txt">Table of n, a(n) for n = 1..1001</a>

%t Select[Range[6000], Mod[If[Abs[#] < 2, 0, # Total[#2/#1 & @@@ FactorInteger[Abs@ #]]], EulerPhi[#]] == 0 &] (* _Michael De Vlieger_, Mar 11 2021 *)

%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 isA342009(n) = !(A003415(n)%eulerphi(n));

%Y Cf. A000010, A003415, A166374 (subsequence), A173557, A342001, A342008.

%Y Positions of ones in A342415.

%K nonn

%O 1,2

%A _Antti Karttunen_, Mar 11 2021