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.

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

Antti Karttunen, Table of n, a(n) for n = 0..65537

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

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

Sequence in context: A366390 A243059 A332845 * A190621 A325314 A322953

Adjacent sequences: A351076 A351077 A351078 * A351080 A351081 A351082

Keyword

nonn

Author

Antti Karttunen, Feb 11 2022

Status

approved