OFFSET
0,2
COMMENTS
Question: Is this sequence monotonic? If monotonic, then it certainly cannot settle to zero, which implies that A293430 is infinite and that there are nonzero terms arbitrary far in A293233.
If there are no zero terms, then in a simple binary tree illustrated below (where the left hand child is obtained as 2*parent, and the right hand child is 1 + 2*parent) there are arbitrary long trajectories starting from 1 that consist squarefree numbers (A005117) only. All numbers k that are in such trajectories are marked as <k> (terms of A293430). a(n) = the number of marked numbers at level n, where level 0 is the root 1, level 1 has nodes 2 and 3, level 2 nodes 5, 6, 7, etc.
<1>
|
.................../ \...................
<2> <3>
4......../ \.......<5> <6>......./ \.......<7>
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
8 9 <10> <11> 12 <13> <14> <15>
16 17 18 19 20 <21> <22> <23> 24 25 <26> 27 28 <29> <30> <31>
etc.
---
FORMULA
EXAMPLE
MATHEMATICA
Table[Count[Range[2^n, (2^(n + 1)) - 1], _?(AllTrue[Table[Floor[#/2^e], {e, 0, n}], SquareFreeQ] &)], {n, 0, 20}] (* Michael De Vlieger, Oct 10 2017 *)
PROG
(PARI)
allocatemem(2^30);
up_to_level = 23;
up_to = (2^(1+up_to_level))-1;
v293233 = vector(up_to);
v293233[1] = 1;
write("b293430.txt", 1, " ", 1);
countsA293230 = 1; kA293430 = 2; for(n=2, up_to, if(!bitand(n, n-1), print1(countsA293230, ", "); countsA293230 = 0); v293233[n] = moebius(n)* v293233[n\2]; if(v293233[n], write("b293430.txt", kA293430, " ", n); kA293430++; countsA293230++)); print1(countsA293230);
(PARI)
\\ Much faster algorithm:
allocatemem(2^30);
next_living_bud_or_zero(n) = if(issquarefree(n), n, 0);
nextA293230generation(tops) = { my(new_tops = vecsort(vector(2*#tops, i, next_living_bud_or_zero((2*tops[(i+1)\2])+(i%2))), , 8)); if(0==new_tops[1], vector(#new_tops-1, i, new_tops[1+i]), new_tops); }
tops_of_tree = [1];
write("b293230.txt", 0, " ", 1);
print1(1, ", ");
for(n=1, 64, tops_of_tree = nextA293230generation(tops_of_tree); write("b293230.txt", n, " ", k = length(tops_of_tree)); print1(k, ", "));
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen and Michael De Vlieger, Oct 10 2017
STATUS
approved