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A293519 Number of surviving (but not bifurcating) odd nodes at generation n in the binary tree of persistently squarefree numbers (see A293230). 6
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 (list; graph; refs; listen; history; text; internal format)
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: A305866 A328406 A257396 * A237582 A097352 A076050

Adjacent sequences:  A293516 A293517 A293518 * A293520 A293521 A293522

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 16 2017

STATUS

approved

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Last modified December 7 05:14 EST 2019. Contains 329839 sequences. (Running on oeis4.)