A351079 - OEIS

%I #15 Feb 12 2022 17:47:36

%S 0,1,2,3,0,5,6,7,0,9,10,11,0,13,14,0,0,17,21,19,0,21,22,23,0,25,0,0,0,

%T 29,31,31,0,33,34,0,0,37,38,0,0,41,42,43,0,0,46,47,0,49,0,0,0,53,0,0,

%U 0,57,58,59,0,61,62,0,0,65,66,67,0,0,70,71,0,73,0,0,0,77,78,79,0,0,82,83,0,85,0,0,0,89

%N a(n) is the largest term encountered on the path from n to 0 when iterating the map x -> x', or 0 if 0 cannot be reached from n (or if n is 0). Here x' is the arithmetic derivative of x, A003415.

%C Question: Is there any good upper bound for ratio a(n)/n? See also comments in A351261.

%H Antti Karttunen, <a href="/A351079/b351079.txt">Table of n, a(n) for n = 0..65537</a>

%F For n > 0, a(n) = 0 if A099307(n) = 0, otherwise a(n) = max(n, a(A003415(n))).

%F a(0) = 0 and a(A099309(n)) = 0 for all n.

%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 zero, 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 largest term is 21, therefore a(18) = 21.

%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 A351079(n) = { my(m=n); while(n>1, n = A003415checked(n); m = max(m,n)); if(n,m); };

%Y Cf. A003415, A099307, A099309, A351078, A351261 [= a(A351255(n))].

%K nonn

%O 0,3

%A _Antti Karttunen_, Feb 11 2022