A335905 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.
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, 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, 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
Offset: 1
Keywords
Examples
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:
9
/ \
6 12
/ \ / \
(4) (8) (16)
thus a(9) = 3.
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:
10
|\
| \
| 12
| /\
|/ \
(8) (16)
thus a(10) = 2.
For n = 9, the numbers encountered are 6, 9, 12, thus a(9) = 3.
For n = 67, the numbers encountered are 48, 60, 66, 67, 68, 72, 96, thus a(67) = 7.
For n = 105, the numbers encountered are 48, 72, 90, 96, 105, 108, 120, 144, 192, thus a(105) = 9.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..65537
Programs
-
PARI
A171462(n) = if(1==n,0,(n-(n/vecmax(factor(n)[, 1])))); A335876(n) = if(1==n,2,(n+(n/vecmax(factor(n)[, 1])))); A209229(n) = (n && !bitand(n,n-1)); 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));