A293520 Number of dying nodes (withering branches) at generation n in the binary tree of persistently squarefree numbers (A293230).

0, 0, 0, 0, 1, 2, 2, 1, 2, 3, 3, 4, 9, 6, 13, 11, 21, 32, 40, 52, 60, 64, 90, 129, 169, 242, 321, 434, 549, 808, 1026, 1395, 1929, 2551, 3405, 4578, 6131, 8275, 11196, 14814, 20198, 26823, 36295, 48840, 65337, 87634, 118138, 158324, 212870, 287014

Offset

0,6

Comments

Provided that A293441 is strictly growing, then certainly a(n) < A293441(n), because only even nodes may die and A293441(n-1) gives the number of even nodes at level n.

Links

Table of n, a(n) for n=0..49.

Formula

a(n) = Sum_{k=(2^n)..(2^(1+n))-1)] abs(A293233(k))*[0 == (A008966(2k)+A008966(1+2k))].

Example

a(4) = 1 because in the binary tree illustrated in A293230, it is the only node 22 at the level 4 that does not generate any new buds as both 2*22 = 44 and 1+(2*22) = 45 are nonsquarefree numbers.

Prog

(PARI)
\\ Compute sequences A293230, A293520, A293521, A293522 at the same time:
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+1)%2))), , 8)); if(0==new_tops[1], vector(#new_tops-1, i, new_tops[1+i]), new_tops); }
write_counts(n, tops) = { my(w=0, s=0, b=0, k); for(i=1, #tops, if((tops[i]%2), if(issquarefree(1+(2*tops[i])), b++, s++), if(issquarefree(1+(2*tops[i])), s++, w++)); ); write("b293520.txt", n, " ", w); write("b293521.txt", n, " ", s); write("b293522.txt", n, " ", b); write("b293230.txt", n, " ", k=length(tops)); print1(k, ", "); }
tops_of_tree = [1];
write("b293230.txt", 0, " ", 1);
write("b293520.txt", 0, " ", 0);
write("b293521.txt", 0, " ", 0);
write("b293522.txt", 0, " ", 1);
print1(1, ", ");
for(n=1, 52, tops_of_tree = nextA293230generation(tops_of_tree); write_counts(n, tops_of_tree); );

Crossrefs

Cf. A293230, A293441, A293521, A293522.

Sequence in context: A172313 A022871 A127218 * A292602 A071444 A232441

Adjacent sequences: A293517 A293518 A293519 * A293521 A293522 A293523

Keyword

nonn

Author

Antti Karttunen, Oct 12 2017

Status

approved