A299351 For x=n, iterate the map x -> Product_{k is a prime dividing x} (k + 1), a(n) is the number of steps to see a repeated term for the first time.

3, 2, 2, 3, 2, 4, 3, 3, 3, 2, 1, 4, 3, 3, 3, 3, 2, 4, 3, 4, 3, 3, 2, 3, 4, 3, 3, 4, 3, 4, 3, 3, 3, 3, 2, 5, 4, 4, 3, 4, 3, 4, 3, 3, 3, 3, 2, 4, 3, 3, 4, 3, 2, 3, 3, 4, 4, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 3, 3, 3, 2, 6, 5, 3, 4, 3, 4, 4, 3, 3, 4, 4, 3, 3, 4, 4, 3

Offset: 2

Author

Lars Blomberg, Feb 07 2018

Keywords

• nonn

Comments

It appears that all n end in the orbit (3,4) or the fixed point 12, verified to n=10^8.

Let p,q,r,... be primes that increased by 1 become a power of 2 (the Mersenne primes, A000668). Then for n = p^a*q^b*r^c*..., a,b,c,...>=1 -> (p+1)*(q+1)*(r+1)... = 2^e, e>=2 -> (2+1)=3.

The case 3^k, k>=2 first yields 4 and then 3: -> (3+1)=4=2^2 -> (2+1)=3.

It appears that these are the only ones entering the orbit (3,4), all other n end in the fixed point 12.

Examples

			For n=2: 2 -> (2+1)=3 -> (3+1)=4=2^2 -> (2+1)=3; 3 is repeated so a(2)=3.
For n=19: 19 -> (19+1)=20=2^2*5 -> (2+1)*(5+1)=18=2*3^2 -> (2+1)*(3+1)=12=2^2*3 -> (2+1)*(3+1)=12; 12 is repeated so a(19)=4.
		

Links

• Lars Blomberg, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A299352.