A326197 - OEIS

%I #9 Aug 24 2019 21:29:19

%S 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,

%T 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,

%U 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

%N 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)).

%C It seems that A000961 gives the positions of zeros.

%H Antti Karttunen, <a href="/A326197/b326197.txt">Table of n, a(n) for n = 1..16384</a>

%H Antti Karttunen, <a href="/A326197/a326197.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

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

%e 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.

%e .

%e    12

%e     |

%e     4

%e     | \

%e     |  2

%e     | /

%e     1

%o (PARI)

%o 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))); };

%o A326196(n) = length(A326196aux(n,Set([])));

%o A326197(n) = (numdiv(n) - A326196(n));

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

%K nonn

%O 1,12

%A _Antti Karttunen_, Aug 24 2019