A293428 -id:A293428 - cp's OEIS frontend

A293517 a(n) = A293518(n) - A293519(n); how many more surviving even nodes than surviving (but not bifurcating) odd nodes there are at generation n in the binary tree of persistently squarefree numbers.

0, 1, 1, 1, 1, 2, 1, 3, 4, 5, 9, 8, 10, 20, 17, 36, 32, 50, 66, 83, 118, 171, 219, 291, 410, 511, 730, 952, 1325, 1665, 2389, 3100, 4147, 5631, 7591, 10093, 13756, 18390, 24540, 33288, 44391, 60052, 80291, 108096, 145226, 194764, 262091, 352096, 473452, 635336, 854332, 1147668

Offset: 0

Antti Karttunen, Oct 16 2017

nonn

As long as there are at least as many surviving even than surviving (but not bifurcating) odd nodes at each generation in the tree of persistently squarefree numbers (see illustration in A293230), this sequence also stays nonnegative, and being also the first differences of A293441 guarantees its monotonicity. If A293441 is monotonic, then A293230 is also, which in turn implies also that A293430 has infinite number of terms and that there will be nonzero terms arbitrary far in A293233.

The surviving children of even vertices are all of the form 4k+1, while the surviving children (those without an odd sibling) of odd vertices are all of the form 4k+2.

Cf. A293230, A293233, A293430, A293518, A293519.
First differences of A293441.
Cf. also A293428.

(define (A293517 n) (- (A293518 n) (A293519 n)))

a(n) = A293518(n) - A293519(n).

a(n) = A293441(1+n) - A293441(n).

A293229 a(0) = 0; and for n > 0, a(n) = a(n-1) + (A008966(4n+3) - A008966(4n+1)).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 3, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 3, 4, 4, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 4, 4, 4, 3, 2, 2, 2, 3, 3, 4, 4, 4, 4, 3, 3, 3, 4, 4, 5, 5, 4, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2

Offset: 0

Antti Karttunen, Oct 12 2017

sign

The sequence indicates about a possible bias (or lack of it) in the distribution of squarefree numbers among the numbers of the form 4k+1 vs. the numbers of the form 4k+3. See A293429 for another version.

The first negative term is a(1702) = -1.

Antti Karttunen, Table of n, a(n) for n = 0..10000
Hans Havermann, Plot of n, a(n) for n = 0..200000
Hans Havermann, Plot of n, a(n) for n = 0..10^7

Cf. A008966, A293428, A293429 (a variant).

up_to = 10000; bias = 0; for(k=0,up_to,bias += (issquarefree((4*k)+3)-issquarefree((4*k)+1)); write("b293229.txt", k, " ", bias));

;; With memoization-macro definec.
(definec (A293229 n) (if (zero? n) n (+ (- (A008966 (+ 3 (* 4 n))) (A008966 (+ 1 (* 4 n)))) (A293229 (- n 1)))))

a(0) = 0; and for n > 0, a(n) = a(n-1) + (A008966(4n+3) - A008966(4n+1)).

cp's OEIS Frontend

A293517 a(n) = A293518(n) - A293519(n); how many more surviving even nodes than surviving (but not bifurcating) odd nodes there are at generation n in the binary tree of persistently squarefree numbers.

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