A299352 -id:A299352 - cp's OEIS frontend

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

Lars Blomberg, Feb 07 2018

nonn

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.

			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.

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

Cf. A299352.

A376299 Fixed points of A008473.

Original entry on oeis.org

1, 4, 90, 120

Offset: 1

Darío Clavijo, Sep 19 2024

These are the numbers such that the sum of the powerfree parts of the divisors of n equals n.
These fixed points appeared in A299352 and the linked Combo Class video. The former notes there are no more fixed points <= 10^8. - Michael S. Branicky, Sep 19 2024
Any further terms are > 10^11. - Lucas A. Brown, Oct 19 2024

Lucas A. Brown, Python program.
Combo Class, The Mysterious Pattern I Found Within Prime Factorizations, Sep 12 2024.

Cf. A008473, A299352.

from sympy import factorint, prod
A008473 = lambda n: prod(sum(pk) for pk in factorint(n).items())
isok = lambda n: A008473(n) == n
print([n for n in range(1, 10**6) if isok(n)])

cp's OEIS Frontend

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.

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