|
| |
|
|
A293230
|
|
a(n) is the number of integers k in range [2^n, (2^(n+1))-1] such that all terms in finite sequence [k, floor(k/2), floor(k/4), floor(k/8), ..., 1] are squarefree.
|
|
12
|
|
|
|
1, 2, 3, 5, 7, 9, 12, 15, 19, 26, 35, 49, 66, 84, 114, 151, 204, 272, 354, 470, 619, 820, 1109, 1499, 2009, 2710, 3631, 4872, 6554, 8831, 11821, 15875, 21364, 28611, 38389, 51611, 69295, 93144, 125290, 168220, 226048, 303727, 408170, 548513, 736900, 990222, 1330212, 1787067, 2401254, 3226802, 4335590, 5825258
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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.
---
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = Sum_{k=2^n..2^(1+n)-1} abs(A293233(k)).
a(n) = A293520(n) + A293521(n) + A293522(n). [sum of number of withering, surviving and bifurcating nodes at each level.]
It seems that lim_{n ->oo} A293441(n+1)/a(n) ~= 0.770... (if it exists) and similarly lim_{n ->oo} a(n+1)/a(n) ~= 1.34...
|
|
|
EXAMPLE
|
In range [2^0 .. (2^1)-1] = [1], all terms (namely 1) are in A293430, thus a(0) = 1.
In range [2^1 .. (2^2)-1] = [2 .. 3] all terms are in A293430, thus a(1) = 2.
In range [2^2 .. (2^3)-1] = [4 .. 7] the terms 5, 6, 7 are in A293430 (because they themselves are squarefree and when applying x -> floor(x/2) to them, give either 2 or 3, numbers that are also included in A293430), thus a(2) = 3.
|
|
|
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
|
Cf. A293440 (the first differences).
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
