A351078 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.

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, 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, 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

Offset: 0

Author

Antti Karttunen, Feb 11 2022

Keywords

• nonn
• look

Comments

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

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

Examples

			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.
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.
		

Links

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

Crossrefs

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

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

Cf. also A327968.

Programs

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

Formula

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

a(p) = p for all primes p.