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

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

Author

Antti Karttunen, Oct 12 2017

Keywords

• sign

Comments

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.

Links

• 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

Crossrefs

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

Programs

PARI

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

Scheme

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

Formula

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