A326197 Number of divisors of n that are not reachable from n with any combination of transitions x -> gcd(x,sigma(x)) and x -> gcd(x,phi(x)).

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 1, 0, 2, 1, 1, 0, 2, 0, 1, 0, 2, 0, 4, 0, 0, 1, 1, 2, 4, 0, 1, 1, 2, 0, 4, 0, 2, 3, 1, 0, 4, 0, 2, 1, 2, 0, 2, 1, 3, 1, 1, 0, 7, 0, 1, 2, 0, 2, 4, 0, 2, 1, 5, 0, 4, 0, 1, 3, 2, 2, 4, 0, 4, 0, 1, 0, 6, 2, 1, 1, 3, 0, 6, 1, 2, 1, 1, 1, 4, 0, 2, 3, 4, 0, 4, 0, 3, 5

Offset: 1

Author

Antti Karttunen, Aug 24 2019

Keywords

• nonn

Comments

It seems that A000961 gives the positions of zeros.

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)). Only the divisors 3 and 6 of 12 are not included in the directed acyclic graph formed from those two transitions (see illustration below), thus a(12) = 2.
.
   12
    |
    4
    | \
    |  2
    | /
    1
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A000005, A000961, A009194, A009195, A326196.

Programs

PARI

A326196aux(n,distvals) = { distvals = setunion([n],distvals); if(1==n,distvals, my(a=gcd(n,eulerphi(n)), b=gcd(n,sigma(n))); distvals = A326196aux(a,distvals); if((a==b)||(b==n),distvals, A326196aux(b,distvals))); };
A326196(n) = length(A326196aux(n,Set([])));
A326197(n) = (numdiv(n) - A326196(n));

Formula

a(n) = A000005(n) - A326196(n).