%I #15 Aug 26 2019 12:51:45

%S 1,2,2,2,2,1,2,2,2,4,2,4,2,5,4,2,2,6,2,5,3,6,2,5,2,4,2,5,2,4,2,2,5,6,

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

%U 4,8,2,7,2,7,3,6,3,3,2,7,2,6,2,7,3,6,6,7,2,1,5,6,4,8,4,9,2,9,5,9,2,4,2,7,7

%N Number of positive integers that are reachable from n with some combination of transitions x -> usigma(x)-x and x -> gcd(x,usigma(x)), where usigma is the sum of unitary divisors of n (A034448).

%C Question: Is this sequence well defined for every n ? If A318882 is not well defined in whole N, then neither this can be.

%H Antti Karttunen, <a href="/A327160/b327160.txt">Table of n, a(n) for n = 1..20000</a>

%H Antti Karttunen, <a href="/A327160/a327160.txt">Data supplement: n, a(n) computed for n = 1..100000</a>

%e From n = 30 we can reach any of the following strictly positive numbers: 30 (e.g., with an empty sequence of transitions), 42 (as A034460(30) = 42), 54 (as A034460(42) = 54; note that A034460(54) = 30 again) and 6 as A323166(30) = A323166(42) = A323166(54) = 6 = A323166(6) = A034460(6), thus a(30) = 4.

%o (PARI)

%o A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448

%o A327160aux(n,xs) = if(vecsearch(xs,n),xs, xs = setunion([n],xs); if(1==n,xs, my(a=A034448(n)-n, b=gcd(A034448(n),n)); xs = A327160aux(a,xs); if((a==b),xs, A327160aux(b,xs))));

%o A327160(n) = length(A327160aux(n,Set([])));

%Y Cf. A034448, A034460, A318882, A323166.

%Y Cf. also A326198, A327161.

%K nonn

%O 1,2

%A _Antti Karttunen_, Aug 25 2019