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 A293520 Number of dying nodes (withering branches) at generation n in the binary tree of persistently squarefree numbers (A293230). 6
 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 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified July 25 09:49 EDT 2021. Contains 346289 sequences. (Running on oeis4.)