A326196 Number of divisors of n that are reachable from n with some combination of transitions x -> gcd(x,sigma(x)) and x -> gcd(x,phi(x)).

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 3, 5, 2, 5, 2, 4, 3, 3, 2, 6, 3, 3, 4, 4, 2, 4, 2, 6, 3, 3, 2, 5, 2, 3, 3, 6, 2, 4, 2, 4, 3, 3, 2, 6, 3, 4, 3, 4, 2, 6, 3, 5, 3, 3, 2, 5, 2, 3, 4, 7, 2, 4, 2, 4, 3, 3, 2, 8, 2, 3, 3, 4, 2, 4, 2, 6, 5, 3, 2, 6, 2, 3, 3, 5, 2, 6, 3, 4, 3, 3, 3, 8, 2, 4, 3, 5, 2, 4, 2, 5, 3

Offset: 1

Author

Antti Karttunen, Aug 24 2019

Keywords

• nonn

Comments

Number of distinct vertices in the directed acyclic graph formed by edge relations x -> A009194(x) and x -> A009195(x), where n is the unique root of the graph.

Because both A009194(n) and A009195(n) are divisors of n, it means that any number reached from n must also be a divisor of n. Number n is also included in the count as it is reached with an empty sequence of transitions.

Question: Are there any other numbers than those in A000961 for which a(n) = A000005(n) ?

Examples

			From n = 12 we can reach any of the following of its 6 divisors: 12 (with an empty combination of transitions), 4 (as A009194(12) = A009195(12) = 4), 2 (as A009195(4) = 2) and 1 (as A009194(4) = 1 = A009194(2) = A009195(2)), thus a(12) = 4.
The directed acyclic graph formed from those two transitions with 12 as its unique root looks like this:
   12
    |
    4
    | \
    |  2
    | /
    1
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000005, A000961, A009194, A009195, A326192, A326194, A326195, A326197.

Cf. also A326189, A326198.

Programs

PARI

A326196aux(n,xs) = { xs = setunion([n],xs); if(1==n,xs, my(a=gcd(n,eulerphi(n)), b=gcd(n,sigma(n))); xs = A326196aux(a,xs); if((a==b)||(b==n),xs, A326196aux(b,xs))); };
A326196(n) = length(A326196aux(n,Set([])));

Formula

a(n) = A000005(n) - A326197(n).

a(n) > max(A326194(n), A326195(n)).