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A359542 Inverse Möbius transform of A341999, which is the characteristic function of numbers that will never reach zero when iterated with the arithmetic derivative.

%I A359542 #10 Jan 05 2023 18:30:52
%S A359542 0,0,0,1,0,0,0,2,0,0,0,2,0,0,1,3,0,0,0,2,0,0,0,4,0,1,1,2,0,1,0,4,0,0,
%T A359542 1,3,0,0,1,4,0,0,0,2,2,0,0,6,0,1,1,3,0,2,1,4,0,0,0,5,0,0,1,5,0,0,0,2,
%U A359542 1,1,0,6,0,1,2,2,0,2,0,6,2,0,0,4,0,1,1,4,0,3,1,2,0,0,1,8,0,0,1,4,0,2,0,5,2
%N A359542 Inverse Möbius transform of A341999, which is the characteristic function of numbers that will never reach zero when iterated with the arithmetic derivative.
%C A359542 Question: Why there seems to be frequency bands (horizontal stripes) in the scatter plot?
%H A359542 Antti Karttunen, <a href="/A359542/b359542.txt">Table of n, a(n) for n = 1..65537</a>
%F A359542 a(n) = Sum_{d|n} A341999(d).
%F A359542 a(n) = A000005(n) - A359541(n).
%o A359542 (PARI)
%o A359542 A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i, 2]>=f[i, 1], return(0), s += f[i, 2]/f[i, 1])); (n*s));
%o A359542 A341999(n) = if(!n,n,while(n>1, n = A003415checked(n)); (!n));
%o A359542 A359542(n) = sumdiv(n,d,A341999(d));
%Y A359542 Cf. A000005, A328308, A341999, A359541, A359543, A359544 (positions of 0's).
%K A359542 nonn,look
%O A359542 1,8
%A A359542 _Antti Karttunen_, Jan 05 2023