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

0,3

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.

Links

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

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.

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 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.

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

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.

Sequence in context: A265513 A140508 A063956 * A128214 A307865 A219695

Adjacent sequences: A351075 A351076 A351077 * A351079 A351080 A351081

Keyword

nonn,look

Author

Antti Karttunen, Feb 11 2022

Status

approved