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

0,7

Links

Table of n, a(n) for n=0..51.

Formula

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

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

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

Example

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.

Prog

(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)))))))

Crossrefs

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

Sequence in context: A346708 A328406 A257396 * A237582 A097352 A076050

Adjacent sequences: A293516 A293517 A293518 * A293520 A293521 A293522

Keyword

nonn

Author

Antti Karttunen, Oct 16 2017

Status

approved