%I #18 Feb 12 2022 17:46:34

%S 0,1,2,3,0,5,5,7,0,5,7,11,0,13,5,0,0,17,7,19,0,7,13,23,0,7,0,0,0,29,

%T 31,31,0,5,19,0,0,37,7,0,0,41,41,43,0,0,7,47,0,5,0,0,0,53,0,0,0,13,31,

%U 59,0,61,5,0,0,7,61,67,0,0,59,71,0,73,0,0,0,7,71,79,0,0,43,83,0,13,0,0,0,89,0,0,0,19,5

%N First noncomposite number reached when iterating the map x -> x', when starting from x = n, or 0 if no such number is ever reached. Here x' is the arithmetic derivative of x, A003415.

%C Primes of A189483 occur only once, on the corresponding indices, while A189441 may also occur in other positions.

%C There are interesting white "filament-like regions" in the scatter plot.

%H Antti Karttunen, <a href="/A351078/b351078.txt">Table of n, a(n) for n = 0..65537</a>

%F For all n, a(4*n) = a(27*n) = a((p^p)*n) = a(A099309(n)) = 0.

%F a(p) = p for all primes p.

%e For n = 15, if we iterate with A003415, we get a path 15 -> 8 -> 12 -> 16 -> 32 -> 80 -> 176 -> 368 -> ..., where the terms just keep on growing without ever reaching a prime or 1, therefore a(15) = 0.

%e For n = 18, its path down to zero, when iterating A003415 is: 18 -> 21 -> 10 -> 7 -> 1 -> 0, and the first noncomposite term on the path is prime 7, therefore a(18) = 7.

%o (PARI)

%o 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 A351078(n) = { while(n>1&&!isprime(n), n = A003415checked(n)); (n); };

%Y Cf. A003415, A189441, A189483, A351079, A351259 [= a(A351255(n))].

%Y Cf. A099309 (positions of zeros after the initial one at a(0)=0), A328115 (positions of 5's), A328117 (positions of 7's).

%Y Cf. also A327968.

%K nonn,look

%O 0,3

%A _Antti Karttunen_, Feb 11 2022