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.

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

Offset: 0

Author

Antti Karttunen and Michael De Vlieger, Oct 10 2017

Keywords

• nonn

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 (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.

---

Examples

			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.
		

Crossrefs

Cf. A005117, A293233, A293430, A293441, A293518, A293519, A293520, A293521, A293522.

Cf. A293440 (the first differences).

Programs

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 *)

PARI

\\ A naive algorithm that computes A293233, A293430 and A293230 at the same time:
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, ", "));

Formula

a(n) = Sum_{k=2^n..2^(1+n)-1} abs(A293233(k)).

For n >= 1, a(n) = A293441(n) + A293441(n-1).

a(n) = A293520(n) + A293521(n) + A293522(n). [sum of number of withering, surviving and bifurcating nodes at each level.]

a(n) = A293520(n) + (A293518(n) + A293519(n)) + A293522(n).

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...