%I #16 Jul 22 2020 13:25:37

%S 0,0,1,0,2,1,2,0,3,2,3,1,3,2,4,0,4,3,5,2,4,3,4,1,6,3,6,2,6,4,5,0,5,4,

%T 6,3,6,5,5,2,6,4,7,3,7,4,5,1,6,6,7,3,9,6,7,2,8,6,7,4,6,5,7,0,7,5,7,4,

%U 6,6,7,3,7,6,9,5,8,5,7,2,10,6,7,4,9,7,9,3,10,7,7,4,8,5,11,1,7,6,8,6,11,7,10,3,9

%N Number of distinct integers encountered on all possible paths from n to any first encountered powers of 2 (that are excluded from the count), when using the transitions x -> x - (x/p) and x -> x + (x/p) in any order, where p is the largest prime dividing x.

%H Antti Karttunen, <a href="/A335905/b335905.txt">Table of n, a(n) for n = 1..65537</a>

%e From 9 one can reach with the transitions x -> A171462(x) (leftward arrow) and x -> A335876(x) (rightward arrow) the following three numbers, when one doesn't expand any power of 2 (in this case, 4, 8 and 16, that are not included in the count) further:

%e 9

%e / \

%e 6 12

%e / \ / \

%e (4) (8) (16)

%e thus a(9) = 3.

%e From 10 one can reach with the transitions x -> A171462(x) and x -> A335876(x) the following two numbers (10 & 12), when one doesn't expand any powers of 2 (8 and 16 in this case, not counted) further:

%e 10

%e |\

%e | \

%e | 12

%e | /\

%e |/ \

%e (8) (16)

%e thus a(10) = 2.

%e For n = 9, the numbers encountered are 6, 9, 12, thus a(9) = 3.

%e For n = 67, the numbers encountered are 48, 60, 66, 67, 68, 72, 96, thus a(67) = 7.

%e For n = 105, the numbers encountered are 48, 72, 90, 96, 105, 108, 120, 144, 192, thus a(105) = 9.

%o (PARI)

%o A171462(n) = if(1==n,0,(n-(n/vecmax(factor(n)[, 1]))));

%o A335876(n) = if(1==n,2,(n+(n/vecmax(factor(n)[, 1]))));

%o A209229(n) = (n && !bitand(n,n-1));

%o A335905(n) = if(A209229(n),0,my(xs=Set([n]),allxs=xs,newxs,a,b,u); for(k=1,oo, newxs=Set([]); if(!#xs, return(#allxs)); allxs = setunion(allxs,xs); for(i=1,#xs,u = xs[i]; a = A171462(u); if(!A209229(a), newxs = setunion([a],newxs)); b = A335876(u); if(!A209229(b), newxs = setunion([b],newxs))); xs = newxs));

%Y Cf. A006530, A171462, A335876, A335906.

%Y Cf. also A332809, A335884, A335885, A335904.

%K nonn

%O 1,5

%A _Antti Karttunen_, Jun 30 2020