A293429 -id:A293429 - cp's OEIS frontend

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

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)).

A293428 a(0) = 1; and for n > 0, a(n) = a(n-1) + (A008966(4n+1) - A107078(4n+3)).

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 4, 5, 6, 7, 8, 8, 8, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 16, 16, 17, 18, 19, 20, 20, 20, 20, 21, 21, 22, 23, 23, 24, 24, 25, 26, 27, 26, 26, 27, 28, 29, 29, 30, 31, 32, 32, 33, 34, 35, 36, 36, 37, 38, 39, 39, 39, 40, 41, 42, 42, 43, 44, 44, 44, 45, 46, 46, 47, 47, 48, 49, 50, 50, 51, 52, 52, 53, 53

Offset: 0

Antti Karttunen, Oct 16 2017

nonn

After a(0)=1, each term a(n) is either (a) one more than its predecessor, in case 4n+1 and 4n+3 are both squarefree, (b) stays same if only the other one is squarefree, or (c) decreases by one if neither 4n+1 nor 4n+3 are squarefree.

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A008966, A107078, A293229, A293429.

Cf. also A293517.

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

Or for n > 0, a(n) = a(n-1) + (A008966(4n+1) - A107078(4n+3)).

cp's OEIS Frontend

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

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