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A351079
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.
6
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, 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, 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
OFFSET
0,3
COMMENTS
Question: Is there any good upper bound for ratio a(n)/n? See also comments in A351261.
LINKS
FORMULA
For n > 0, a(n) = 0 if A099307(n) = 0, otherwise a(n) = max(n, a(A003415(n))).
a(0) = 0 and a(A099309(n)) = 0 for all n.
EXAMPLE
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.
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.
PROG
(PARI)
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));
A351079(n) = { my(m=n); while(n>1, n = A003415checked(n); m = max(m, n)); if(n, m); };
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Feb 11 2022
STATUS
approved