A293519 Number of surviving (but not bifurcating) odd nodes at generation n in the binary tree of persistently squarefree numbers (see A293230).

0, 0, 0, 1, 1, 0, 2, 3, 2, 3, 3, 8, 10, 11, 17, 20, 31, 38, 46, 67, 90, 116, 160, 220, 280, 397, 509, 685, 927, 1280, 1663, 2248, 3056, 4050, 5383, 7339, 9714, 13029, 17714, 23738, 31791, 42793, 57473, 77175, 103839, 140100, 187495, 252068, 338257, 454325, 611101, 820924

Offset: 0

Author

Antti Karttunen, Oct 16 2017

Keywords

• nonn

Examples

			a(3) = 1 because in the binary tree illustrated in A293230, there is only one odd node at the level 3 (namely, the node 13) that spawns just one offspring.
		

Crossrefs

Cf. A008966, A293230, A293233, A293517, A293518, A293521.

Programs

PARI

\\ A naive algorithm (see A293518 for a better program):
up_to_level = 28;
up_to = (2^(1+up_to_level));
is_persistently_squarefree(n,base) = { while(n>1, if(!issquarefree(n),return(0)); n \= base); (1); };
{ countsA293441 = 1; countsA293519 = 0; k=1; n=3; while(n <= 1+up_to,if(!bitand(n-1,n-2), write("b293441.txt", k, " ", countsA293441); write("b293519.txt", k, " ", countsA293519); print1(countsA293519,", "); countsA293441 = 0; countsA293519 = 0; k++); if(is_persistently_squarefree(n,2),countsA293441++; if(!issquarefree(1+(2*n)),countsA293519++)); n += 2); }

Scheme

(define (A293519 n) (add (lambda (k) (* (if (and (= 1 (A008966 (+ k k))) (= 0 (A008966 (+ 1 k k)))) 1 0) (abs (A293233 k)))) (A000079 n) (+ -1 (A000079 (+ 1 n)))))
;; Implements sum_{i=lowlim..uplim} intfun(i)
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

Formula

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

A293518(n) + a(n) = A293521(n).

A293518(n) - a(n) = A293517(n).